hcju/isabelleps huggingface dataset:
WTrtCall: fixes P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and sh :: "sheap"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and C :: "char list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and pns :: "char list list"   and body :: "char list exp"   and D :: "char list"   and es :: "char list exp list"   and Ts' :: "ty list" assumes "P,E,h,sh \<turnstile> e : Class C"   and "P \<turnstile> C sees M, NonStatic :  Ts\<rightarrow>T = (pns, body) in D"   and "P,E,h,sh \<turnstile> es [:] Ts'"   and "P \<turnstile> Ts' [\<le>] Ts" shows "P,E,h,sh \<turnstile> e\<bullet>M(es) : T"
mult_isol: fixes x :: "'b"   and y :: "'b"   and z :: "'b" assumes "x \<le> y" shows "z \<cdot> x \<le> z \<cdot> y"
1: fixes v :: "V"   and u :: "V" assumes "v \<in> elts u"   and "u \<in> elts x" shows "v \<in> elts (TC x)"
absolutely_integrable_component: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and b :: "'b" assumes "f absolutely_integrable_on A" shows "(\<lambda>x. f x \<bullet> b) absolutely_integrable_on A"
arith_mod_nzero: fixes i :: "nat"   and n :: "nat"   and t :: "nat" assumes "i < n"   and "0 < i" shows "0 < (n * t + i) mod n"
assms_3: fixes M :: "nat \<Rightarrow> 'a" assumes "decseq M"   and "\<And>i. P (M i)" shows "P (Inf (range M))"
card_of_min_alg2: fixes s1 :: "('e, 'f, 'd) F1 \<Rightarrow> 'f"   and s2 :: "('e, 'f, 'd) F2 \<Rightarrow> 'd" shows "|min_alg2 s1 s2| \<le>o ASucFbd"
im_sheaf_def: 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"   and S' :: "'f set"   and is_open' :: "'f set \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'f"   and V :: "'f set" assumes "im_sheaf S is_open \<FF> \<rho> b add_str mult_str zero_str one_str S' is_open' f" shows "im_sheaf.im_sheaf S \<FF> f V \<equiv> \<FF> (f \<^sup>\<inverse> S V)"
asIMOD_igSubst: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'c) semDom"   and ys :: "'varSort"   and Y :: "('varSort \<Rightarrow> 'var \<Rightarrow> 'c) \<Rightarrow> 'c"   and y :: "'var"   and X :: "('varSort \<Rightarrow> 'var \<Rightarrow> 'c) \<Rightarrow> 'c" shows "igSubst (asIMOD SEM) ys Y y X = (\<lambda>val. if sWlsVal SEM val then X (val (y := Y val)_ys) else undefined)"
CD_eq: fixes K :: "nat"   and X :: "msg" shows "Crypt K (Decrypt K X) = X"
Fn_def: shows "Fn \<equiv> map_fun id (map_fun id (map_fun rep_trm abs_trm)) PFn"
attach_converter_of_resource_conv_parallel_resource: fixes res :: "('h, 'j) resource"   and res' :: "('i, 'k) resource" shows "converter_of_resource res |\<^sub>\<propto> 1\<^sub>C \<rhd> res' = res \<parallel> res'"
not_Cons_self2: fixes x :: "'a"   and xs :: "'a list" shows "x # xs \<noteq> xs"
Block\<^sub>1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and e :: "nat exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and e' :: "nat exp"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and i :: "nat"   and T :: "ty" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>e',s\<^sub>1\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>{i:T; e},s\<^sub>0\<rangle> \<Rightarrow> \<langle>e',s\<^sub>1\<rangle>"
assms_2: shows "r \<down>s S \<subseteq> S"
the_eqI: fixes P :: "'b Predicate.pred"   and x :: "'b" assumes "(THE x. pred.eval P x) = x" shows "Predicate.the P = x"
composable: shows "\<mu>.cod.src = \<nu>.cod.trg"
r_univuniv2_recfn: shows "recfn 3 r_univuniv2"
Pderiv_PDERIVS: fixes R' :: "'b rexp set"   and R :: "'b rexp set"   and a :: "'b" assumes "R' \<subseteq> PDERIVS R" shows "Pderiv a R' \<subseteq> PDERIVS R"
mod_if: fixes m :: "nat"   and n :: "nat" shows "m mod n = (if m < n then m else (m - n) mod n)"
simps_1: fixes sa :: "'b"   and f :: "'e \<rightarrow> ('c, 'd) Stream"   and ha :: "'b \<rightarrow> ('e, 'b) Step" assumes "sa \<noteq> \<bottom>" shows "concatMapStep\<cdot>f\<cdot>ha\<cdot>(sa :!: Nothing) = (case ha\<cdot>sa of Done \<Rightarrow> Done | Skip\<cdot>sa' \<Rightarrow> Skip\<cdot>(sa' :!: Nothing) | Yield\<cdot>a\<cdot>sa' \<Rightarrow> Skip\<cdot>(sa' :!: Just\<cdot>(f\<cdot>a)))"
inner_verts_def: fixes p :: "('a \<times> 'a) list" shows "H.inner_verts p \<equiv> tl (map fst p)"
proves_eq_reflexive_pre: fixes \<Theta> :: "((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 t :: "Core.term" assumes "wf_theory \<Theta>"   and "term_ok \<Theta> t" shows "\<Theta>,{} \<turnstile> mk_eq' (the (typ_of t)) t t"
get_operand2_def: fixes op_list :: "inst_operand list"   and s :: "'b sparc_state" shows "get_operand2 op_list s \<equiv> let flagi = get_operand_flag (op_list ! 0); curr_win = ucast (get_CWP (cpu_reg_val PSR s)) in if flagi = 0 then let rs2 = get_operand_w5 (op_list ! 2); rs2_val = user_reg_val curr_win rs2 s in rs2_val else let ext_simm13 = sign_ext13 (get_operand_simm13 (op_list ! 2)) in ext_simm13"
hom_closed: fixes x :: "'a" assumes "x \<in> carrier G" shows "h x \<in> carrier (H\<lparr>carrier := h ` carrier G\<rparr>)"
countable_subset_image: fixes B :: "'c set"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "(countable B \<and> B \<subseteq> f ` A) = (\<exists>A'. countable A' \<and> A' \<subseteq> A \<and> B = f ` A')"
good1: shows "is_interp (adjointFO I \<sigma> \<nu>)"
image_set: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "f ` set xs = set (map f xs)"
diff_add_eq: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "class.ab_group_add plus zero minus uminus" shows "plus (minus a b) c = minus (plus a c) b"
open_inter: fixes U :: "'a set"   and V :: "'a set" assumes "is_open U"   and "is_open V" shows "is_open (U \<inter> V)"
lift_trimmed: fixes C :: "'a"   and e :: "real"   and xs :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real" assumes "trimmed C" shows "trimmed (MS (MSLCons (C, e) xs) f)"
iso_runit: fixes a :: "'c local.term set" assumes "\<F>C.ide a" shows "\<F>C.iso (\<F>C.runit a)"
distinct_17: fixes x31 :: "nat"   and x32 :: "'a"   and x33 :: "'a"   and x34 :: "'b"   and x35 :: "'b"   and x36 :: "'c"   and x51 :: "nat"   and x52 :: "'a"   and x53 :: "'a" shows "Recv x31 x32 x33 x34 x35 x36 \<noteq> RecvMarker x51 x52 x53"
filter_eq_ConsD: fixes P :: "'b \<Rightarrow> bool"   and ys :: "'b list"   and x :: "'b"   and xs :: "'b list" assumes "filter P ys = x # xs" shows "\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs"
fk: fixes y :: "'b" assumes "y \<in> T" shows "f (k y) = y"
meas: fixes Ab :: "'a set"   and tb :: "real" assumes "finite Ab" shows "S Ab tb \<inter> space (Pi\<^sub>M Ab (\<lambda>_. lborel)) \<in> sets (Pi\<^sub>M Ab (\<lambda>_. lborel))"
prems_1: fixes xt :: "'a \<times> 'b" assumes "xt \<in> S \<times> cbox a b" shows "(\<lambda>(x, y). f x y) differentiable at xt"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>A\<^sub>1) (\<cdot>\<^sub>A\<^sub>1) A1.map A1.map A1.map"
inf_sup_principal_aux: fixes F :: "'a set"   and G :: "'a set" assumes "ord_class.filter F"   and "ord_class.filter G"   and "is_principal_up (filter_sup F G)"   and "is_principal_up (F \<inter> G)" shows "is_principal_up F"
inject: fixes x1 :: "'a set"   and x2 :: "'a set"   and y1 :: "'a set"   and y2 :: "'a set" shows "(Kripke x1 x2 = Kripke y1 y2) = (x1 = y1 \<and> x2 = y2)"
\<Sigma>\<^sub>f_unfold: shows "\<Union> (range \<Sigma>n) - \<C> = {f. 0 < arity f}"
br_chain: fixes \<beta> :: "'a \<Rightarrow> 'd"   and J :: "'a \<Rightarrow> bool"   and \<alpha> :: "'d \<Rightarrow> 'c"   and I :: "'d \<Rightarrow> bool" shows "br \<beta> J O br \<alpha> I = br (\<alpha> \<circ> \<beta>) (\<lambda>s. J s \<and> I (\<beta> s))"
holo1: shows "(\<lambda>z. complex_interval_lebesgue_integral lborel 0 (ereal (exp (- 1))) (f z)) holomorphic_on {z. - 2 < Re z}"
simps_3: fixes uu :: "'d"   and pr :: "('c, 'd) Digit"   and m :: "('c, 'd) FingerTreeStruc"   and sf :: "('c, 'd) Digit" shows "toList (Deep uu pr m sf) = digitToList pr @ toList m @ digitToList sf"
proves_eq_combination_pre: fixes \<Theta> :: "((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 f :: "Core.term"   and \<tau> :: "typ"   and \<tau>' :: "typ"   and g :: "Core.term"   and x :: "Core.term"   and y :: "Core.term" assumes "wf_theory \<Theta>"   and "term_ok \<Theta> f"   and "typ_of f = Some (\<tau> \<rightarrow> \<tau>')"   and "term_ok \<Theta> g"   and "typ_of g = Some (\<tau> \<rightarrow> \<tau>')"   and "term_ok \<Theta> x"   and "typ_of x = Some \<tau>"   and "term_ok \<Theta> y"   and "typ_of y = Some \<tau>" shows "\<Theta>,{} \<turnstile> mk_eq f g \<longmapsto> mk_eq x y \<longmapsto> mk_eq (f $ x) (g $ y)"
face_of_subset_rel_boundary: fixes T :: "'b set"   and S :: "'b set" assumes "T face_of S"   and "T \<noteq> S" shows "T \<subseteq> S - rel_interior S"
mult_mat_vec_def: fixes A :: "'b mat"   and v :: "'b vec" shows "A *\<^sub>v v \<equiv> vec (dim_row A) (\<lambda>i. row A i \<bullet> v)"
semiring_ow_axioms_def: fixes U :: "'c set"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "semiring_ow_axioms U plus times \<equiv> (\<forall>a b c. a \<in> U \<longrightarrow> b \<in> U \<longrightarrow> c \<in> U \<longrightarrow> times (plus a b) c = plus (times a c) (times b c)) \<and> (\<forall>a b c. a \<in> U \<longrightarrow> b \<in> U \<longrightarrow> c \<in> U \<longrightarrow> times a (plus b c) = plus (times a b) (times a c))"
compl: fixes carrier :: "'b set"   and relation :: "('b \<times> 'b) set" assumes "rational_preference carrier relation" shows "\<forall>x\<in>carrier. \<forall>y\<in>carrier. x \<succeq>[relation] y \<or> y \<succeq>[relation] x"
ok_I_def: fixes \<AA> :: "interp"   and fs :: "'a list" shows "ok_I \<AA> fs = (\<forall>x\<in>set fs. index fs x\<^bsup>\<AA>\<^esup>FO \<noteq> {||})"
inner_image_box: assumes "\<i> \<in> Basis"   and "\<forall>i\<in>Basis. a \<bullet> i < b \<bullet> i" shows "(\<lambda>x. x \<bullet> \<i>) ` box a b = {a \<bullet> \<i><..<b \<bullet> \<i>}"
\<delta>\<^sub>p_def_2: shows "(\<delta>\<^sub>p', p\<^sub>1') = find_closest_bf_code p\<^sub>0 (p\<^sub>2 # p\<^sub>3 # ps)"
prefix_llist_of_inf_llist: fixes u :: "'a llist"   and v :: "'a llist" assumes "lfinite u"   and "u \<le> v" shows "list_of u \<le>\<^sub>F\<^sub>I stream_of_llist v"
out_to_bet: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A' :: "'a"   and B' :: "'a"   and C' :: "'a"   and B :: "'a"   and A :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Col Bet A' B' C'"   and "Tarski_neutral_dimensionless.Out Bet B A C = Tarski_neutral_dimensionless.Out Bet B' A' C'"   and "Bet A B C" shows "Bet A' B' C'"
comp_Abs_inverse: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'a \<Rightarrow> 'c" assumes "f \<circ> Abs = g" shows "g \<circ> Rep = f"
LinkMark_correct: shows "Hoare_dgr (R'_a .. (R_a .. SetMarkInv)) LinkMark ((R'_a .. (R_a .. SetMarkInv)) \<sqinter> - grd (step LinkMark))"
emeasure_pair_measure_Times: fixes A :: "'a set"   and N :: "'a measure"   and B :: "'b set" assumes "A \<in> sets N"   and "B \<in> sets lborel" shows "emeasure (N \<Otimes>\<^sub>M lborel) (A \<times> B) = emeasure N A * emeasure lborel B"
tendsto_infdist: fixes f :: "'c \<Rightarrow> 'd"   and l :: "'d"   and F :: "'c filter"   and A :: "'d set" assumes "(f \<longlongrightarrow> l) F" shows "((\<lambda>x. infdist (f x) A) \<longlongrightarrow> infdist l A) F"
ClassD: fixes b :: "'a"   and a :: "'a" assumes "b \<in> Class a"   and "a \<in> S" shows "(b, a) \<in> E"
wf: shows "is_density_expr ([], [], \<Gamma>, CReal 1) t f"
poly_of_vars_def: fixes xs :: "char list multiset" shows "poly_of_vars xs = fold_mset (\<lambda>a. (*) (Var (\<phi> a))) 1 xs"
b: shows "b = monomial (1::'b) (term_of_pair (0::'a, component_of_term u))"
freshAbs_vsubstAbs_E1: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and zs :: "'varSort"   and z :: "'var"   and ys :: "'varSort"   and y1 :: "'var"   and y :: "'var" assumes "goodAbs A"   and "freshAbs zs z (A $[y1 // y]_ys)"   and "zs \<noteq> ys \<or> z \<noteq> y" shows "freshAbs zs z A"
lessThan_Int_lessThan: fixes a :: "'a"   and b :: "'a" shows "{a<..} \<inter> {b<..} = {max a b<..}"
i_aggregate_i_take_mult: fixes k :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'b"   and ag :: "'b list \<Rightarrow> 'b" assumes "0 < k" shows "f_aggregate (f \<Down> (n * k)) k ag = i_aggregate f k ag \<Down> n"
map_of_eq_dom: fixes xs :: "('c \<times> 'd) list"   and ys :: "('c \<times> 'd) list" assumes "map_of xs = map_of ys" shows "fst ` set xs = fst ` set ys"
iffs: fixes nm_name :: "char list"   and nm_receiver_violation :: "bool"   and nm_default :: "'d"   and nm_sinvar :: "'c list_graph \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool"   and nm_offending_flows :: "'c list_graph \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> ('c \<times> 'c) list list"   and nm_node_props :: "('c, 'd) TopoS_Params \<Rightarrow> 'c \<Rightarrow> 'd"   and nm_eval :: "'c list_graph \<Rightarrow> ('c, 'd) TopoS_Params \<Rightarrow> bool"   and more :: "'e"   and nm_name' :: "char list"   and nm_receiver_violation' :: "bool"   and nm_default' :: "'d"   and nm_sinvar' :: "'c list_graph \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool"   and nm_offending_flows' :: "'c list_graph \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> ('c \<times> 'c) list list"   and nm_node_props' :: "('c, 'd) TopoS_Params \<Rightarrow> 'c \<Rightarrow> 'd"   and nm_eval' :: "'c list_graph \<Rightarrow> ('c, 'd) TopoS_Params \<Rightarrow> bool"   and more' :: "'e" shows "(\<lparr>nm_name = nm_name, nm_receiver_violation = nm_receiver_violation, nm_default = nm_default, nm_sinvar = nm_sinvar, nm_offending_flows = nm_offending_flows, nm_node_props = nm_node_props, nm_eval = nm_eval, \<dots> = more\<rparr> = \<lparr>nm_name = nm_name', nm_receiver_violation = nm_receiver_violation', nm_default = nm_default', nm_sinvar = nm_sinvar', nm_offending_flows = nm_offending_flows', nm_node_props = nm_node_props', nm_eval = nm_eval', \<dots> = more'\<rparr>) = (nm_name = nm_name' \<and> nm_receiver_violation = nm_receiver_violation' \<and> nm_default = nm_default' \<and> nm_sinvar = nm_sinvar' \<and> nm_offending_flows = nm_offending_flows' \<and> nm_node_props = nm_node_props' \<and> nm_eval = nm_eval' \<and> more = more')"
prime_elem_def: fixes p :: "'a" shows "prime_elem p = (p \<noteq> (0::'a) \<and> \<not> is_unit p \<and> (\<forall>a b. p dvd a * b \<longrightarrow> p dvd a \<or> p dvd b))"
Quotient_lmap_Abs_Rep: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and Abs :: "'e \<Rightarrow> 'f"   and Rep :: "'f \<Rightarrow> 'e"   and a :: "'f llist" assumes "Quotient3 R Abs Rep" shows "lmap Abs (lmap Rep a) = a"
inject: fixes x1 :: "'l"   and x2 :: "'m"   and x3 :: "'n list"   and y1 :: "'l"   and y2 :: "'m"   and y3 :: "'n list" shows "(State x1 x2 x3 = State y1 y2 y3) = (x1 = y1 \<and> x2 = y2 \<and> x3 = y3)"
alphaAbs_qAbs_iff_ex_qFresh: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and xs :: "'varSort"   and x :: "'var"   and xs' :: "'varSort"   and x' :: "'var"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm" assumes "qGood X" shows "(qAbs xs x X $= qAbs xs' x' X') = (xs = xs' \<and> (\<exists>y. qFresh xs y X \<and> qFresh xs y X' \<and> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs))"
CallThrowParams: fixes es :: "char list exp list"   and vs :: "val list"   and e :: "char list exp"   and es' :: "char list exp list"   and v :: "val"   and M :: "char list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list" assumes "es = map Val vs @ throw e # es'" shows "P \<turnstile> \<langle>Val v\<bullet>M(es),s\<rangle> \<rightarrow> \<langle>throw e,s\<rangle>"
wls_castAbs: fixes xs :: "'varSort"   and s :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set" assumes "wlsAbs (xs, s) A" shows "wls s (snd (castAbs xs s A))"
multi_member_this: fixes x :: "'a"   and XS :: "'a multiset" shows "x \<in># {#x#} + XS"
precedes_def: fixes a :: "'a"   and b :: "'a"   and l :: "'a list" shows "precedes a b l \<equiv> a \<in> set l \<and> b \<in> set l \<and> firstPos a l \<le> firstPos b l"
typeI_cube_line_integral_exists_boundary: shows "\<forall>(k, \<gamma>)\<in>two_chain_boundary two_chain_typeI. line_integral_exists F {i} \<gamma>"
matrix_sum_minus_distrib: fixes n :: "nat"   and f :: "nat \<Rightarrow> complex mat"   and d :: "nat"   and g :: "nat \<Rightarrow> complex mat" assumes "\<And>k. k < n \<Longrightarrow> f k \<in> carrier_mat d d"   and "\<And>k. k < n \<Longrightarrow> g k \<in> carrier_mat d d" shows "matrix_sum d (\<lambda>k. f k - g k) n = matrix_sum d f n - matrix_sum d g n"
ring_of_sets_Pow: fixes sp :: "'b set" shows "ring_of_sets sp (Pow sp)"
vfield_vempty: shows "\<F>\<^sub>\<circ> 0 = 0"
l_get_shadow_root_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_get_shadow_root type_wf get_shadow_root get_shadow_root_locs \<equiv> ((\<forall>element_ptr h h'. reads (get_shadow_root_locs element_ptr) (get_shadow_root element_ptr) h h') \<and> (\<forall>h element_ptr. type_wf h \<longrightarrow> element_ptr |\<in>| element_ptr_kinds h \<longrightarrow> h \<turnstile> ok get_shadow_root element_ptr)) \<and> (\<forall>h element_ptr. h \<turnstile> ok get_shadow_root element_ptr \<longrightarrow> element_ptr |\<in>| element_ptr_kinds h) \<and> (\<forall>element_ptr h. pure (get_shadow_root element_ptr) h)"
poly_of_vec_def: fixes v :: "'a vec" shows "poly_of_vec v \<equiv> let d = dim_vec v in \<Sum>i<d. monom (v $ (d - Suc i)) i"
hyperpow_mult: fixes r :: "'a star"   and s :: "'a star"   and n :: "nat star" shows "(r * s) pow n = r pow n * s pow n"
icd_no_cd_path_shift: fixes i :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n" assumes "i icd\<^bsup>\<pi>\<^esup>\<rightarrow> 0" shows "\<forall>k. \<not> i - 1 cd\<^bsup>\<pi> \<guillemotleft> 1\<^esup>\<rightarrow> k"
lookup_def: fixes xs :: "'x" shows "lookup xs = lookup_raw (list_of_oalistx xs)"
measure_space_inter: fixes s :: "'a set"   and t :: "'a set" assumes "s \<in> sets M"   and "t \<in> sets M"   and "Sigma_Algebra.measure M t = Sigma_Algebra.measure M (space M)" shows "Sigma_Algebra.measure M (s \<inter> t) = Sigma_Algebra.measure M s"
is_rcoset_replistD_cosets: fixes G :: "'a set"   and H :: "'a set"   and gs :: "'a list" assumes "Group G"   and "Group.is_rcoset_replist G H gs" shows "G = (\<Union>g\<in>set gs. Group.rcoset_rel G H `` {g})"
isovar_greater_degree: fixes p :: "'a mpoly"   and var :: "nat" shows "\<forall>i>MPoly_Type.degree p var. isolate_variable_sparse p var i = 0"
appPutField: fixes F :: "char list"   and C :: "char list"   and P :: "'a prog"   and pc :: "nat"   and mxs :: "nat"   and T\<^sub>r :: "ty"   and s :: "ty list \<times> ty err list" shows "app\<^sub>i (Putfield F C, P, pc, mxs, T\<^sub>r, s) = (\<exists>vT vT' oT ST LT. s = (vT # oT # ST, LT) \<and> P \<turnstile> C sees F,NonStatic:vT' in C \<and> subtype P oT (Class C) \<and> subtype P vT vT')"
word_and_not: fixes x :: "'a word" shows "x AND NOT x = 0"
card_1_singleton_iff: fixes A :: "'c set" shows "(card A = Suc 0) = (\<exists>x. A = {x})"
integrable_on_refl: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a" shows "f integrable_on cbox a a"
cases: fixes a :: "('a, 'b, 'c) formula" assumes "fsafe a" obtains t1 :: "('a, 'c) trm" and t2 :: "('a, 'c) trm" where "a = Geq t1 t2" and "dsafe t1" and "dsafe t2"   | args :: "'c \<Rightarrow> ('a, 'c) trm" and p :: "'c" where "a = $\<phi> p__ args" and "\<And>i. dsafe (args i)"   | p :: "('a, 'b, 'c) formula" where "a = ! pa__" and "fsafe pa__"   | p :: "('a, 'b, 'c) formula" and q :: "('a, 'b, 'c) formula" where "a = (pb__ && q)" and "fsafe pb__" and "fsafe q"   | p :: "('a, 'b, 'c) formula" and x :: "'c" where "a = Exists x pc__" and "fsafe pc__"   | aa :: "('a, 'b, 'c) hp" and p :: "('a, 'b, 'c) formula" where "a = (\<langle> aa \<rangle> p)" and "hpsafe aa" and "fsafe p"   | f :: "('a, 'b, 'c) formula" and C :: "'b" where "a = InContext C f" and "fsafe f"
\<delta>\<^sub>0_def_1: shows "\<delta>\<^sub>0 = dist (case p of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))"
assms_10: fixes s2 :: "'s2" shows "\<I> \<turnstile>c oracle2 s2 \<surd>"
not_eq_extremum: fixes a :: "'k" shows "(a \<noteq> \<top>) = (a < \<top>)"
r: fixes l :: "'n" assumes "(\<pi> m, l) \<in> pdg"   and "(l, \<pi> n) \<in> pdg" shows "thesis"
emb_part: fixes s :: "char list \<Rightarrow> int"   and q :: "char list \<Rightarrow> int" shows "Partial_Evaluation.emb (part s) q = s"
bnds_exp_horner: fixes x :: "float"   and prec :: "nat"   and n :: "nat" assumes "real_of_float x \<le> 0" shows "exp (real_of_float x) \<in> {real_of_float (lb_exp_horner prec (get_even n) 1 1 x)..real_of_float (ub_exp_horner prec (get_odd n) 1 1 x)}"
hom: shows "subtopology (Euclidean_space n) S homeomorphic_space subtopology (Euclidean_space n) T"
typing_cond_def: fixes \<A> :: "('fun, 'var) strand_step list" shows "typing_cond \<A> \<equiv> wf\<^sub>s\<^sub>t {} \<A> \<and> fv\<^sub>s\<^sub>t \<A> \<inter> bvars\<^sub>s\<^sub>t \<A> = {} \<and> tfr\<^sub>s\<^sub>t \<A> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (trms\<^sub>s\<^sub>t \<A>) \<and> Ana_invar_subst (ik\<^sub>s\<^sub>t \<A> \<union> assignment_rhs\<^sub>s\<^sub>t \<A>)"
less_eq_Max_atms_of: fixes C :: "'a literal multiset"   and D :: "'a literal multiset" assumes "C \<noteq> {#}"   and "C \<le> D" shows "Max (atms_of C) \<le> Max (atms_of D)"
IH_2: fixes ps :: "(real, 'a) vec list" assumes "set_kdt r \<union> set ps - set (nearest_nbors n ps p r) \<noteq> {}" shows "length (nearest_nbors n ps p r) = n"
implConfid_synth_analz: fixes H :: "msg set"   and A :: "agent"   and B :: "agent"   and M :: "msg" assumes "H \<subseteq> payload \<union> sym.valid \<union> range LtK \<union> Tags"   and "sym.implConfid A B M \<in> synth (analz H)" shows "sym.implConfid A B M \<in> H \<or> M \<in> synth (analz H)"
osum_Partial_order: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" assumes "Partial_order r"   and "Partial_order r'" shows "Partial_order (r +o r')"
general_product_intro: fixes x :: "i"   and F :: "i \<Rightarrow> bool" assumes "\<forall>y. O y x = (\<exists>z. (\<forall>y. F y \<longrightarrow> P z y) \<and> O y z)" shows "(\<pi>x. F x) = x"
M: fixes x :: "'a" shows "Q1 x * \<up> b \<or>\<^sub>A Q2 x * \<up> (\<not> b) \<Longrightarrow>\<^sub>A Q x"