hcju/isabelleps huggingface dataset:
binary_product_of_products_is_product: fixes J0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D0 :: "'b \<Rightarrow> 'a"   and a0 :: "'a"   and J1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D1 :: "'b \<Rightarrow> 'a"   and a1 :: "'a"   and p0 :: "'a"   and p1 :: "'a" assumes "has_as_product J0 D0 a0"   and "has_as_product J1 D1 a1"   and "has_as_binary_product a0 a1 p0 p1"   and "Collect (partial_magma.arr J0) \<inter> Collect (partial_magma.arr J1) = {}"   and "partial_magma.null J0 = partial_magma.null J1" shows "has_as_product (DiscreteCategory.discrete_category.comp (Collect (partial_magma.arr J0) \<union> Collect (partial_magma.arr J1)) (partial_magma.null J0)) (\<lambda>i. if i \<in> Collect (partial_magma.arr J0) then D0 i else if i \<in> Collect (partial_magma.arr J1) then D1 i else null) (local.dom p0)"
a_minus_def: fixes R :: "('e, 'f) ring_scheme"   and x :: "'e"   and y :: "'e" shows "x \<ominus>\<^bsub>R\<^esub> y = x \<oplus>\<^bsub>R\<^esub> \<ominus>\<^bsub>R\<^esub> y"
exp_inv_limit_0_right_gen: fixes k :: "nat" shows "((\<lambda>t. exp (- inverse t) / t ^ k) \<longlongrightarrow> 0) (at_right 0)"
simps_8: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'b BinTree \<Rightarrow> 'b BinTree \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x21 :: "'b BinTree"   and x22 :: "'b BinTree" shows "rec_BinTree f1 f2 (BinTree.Node x21 x22) = f2 x21 x22 (rec_BinTree f1 f2 x21) (rec_BinTree f1 f2 x22)"
B_def: shows "B = (\<lambda>f. box (a f) (b f)) ` (Basis \<rightarrow>\<^sub>E \<rat> \<times> \<rat>)"
prod_z_x1: shows "poly (prod_list_var qs) x1 = 0"
degree_Bernstein: fixes j :: "nat"   and p :: "nat" assumes "j \<le> p" shows "degree (Bernstein_Poly_01 j p) = p"
simps_1: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and e :: "expr" shows "(P,E,h \<turnstile> e :\<^bsub>NT\<^esub> Void) = (P,E,h \<turnstile> e : Void)"
Icc_eq_Icc: fixes l :: "'b word"   and h :: "'b word"   and l' :: "'b word"   and h' :: "'b word" shows "(signed.atLeastAtMost l h = signed.atLeastAtMost l' h') = (l = l' \<and> h = h' \<or> \<not> l \<le>s h \<and> \<not> l' \<le>s h')"
product_concat_map: fixes xs :: "'a list"   and ys :: "'b list" shows "List.product xs ys = concat (map (\<lambda>x. map (Pair x) ys) xs)"
ground_clauses_and_ground_substs: fixes C :: "'b literal set"   and \<sigma> :: "('b \<times> 'b trm) list" assumes "ground_clause (subst_cl C \<sigma>)" shows "ground_on (vars_of_cl C) \<sigma>"
cong_pow: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and 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 modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and division_segment :: "'c \<Rightarrow> 'c"   and b :: "'c"   and c :: "'c"   and a :: "'c"   and n :: "nat" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment"   and "unique_euclidean_semiring.cong modulo b c a" shows "unique_euclidean_semiring.cong modulo (power.power one times b n) (power.power one times c n) a"
isl_def: fixes sum :: "'m + 'n" shows "isl sum = (\<exists>x1. sum = Inl x1)"
hcomplex_HFinite_iff: fixes x :: "complex star" shows "(x \<in> HFinite) = (hRe x \<in> HFinite \<and> hIm x \<in> HFinite)"
upper_lower_hemicontinuous_explicit: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b set"   and T :: "'b set"   and x :: "'a"   and e :: "real" assumes "\<And>x. x \<in> S \<Longrightarrow> f x \<subseteq> T"   and "\<And>U. openin (top_of_set T) U \<Longrightarrow> openin (top_of_set S) {x \<in> S. f x \<subseteq> U}"   and "\<And>U. closedin (top_of_set T) U \<Longrightarrow> closedin (top_of_set S) {x \<in> S. f x \<subseteq> U}"   and "x \<in> S"   and "0 < e"   and "bounded (f x)"   and "f x \<noteq> {}" obtains d :: "real" where "0 < d" and "\<And>x'. \<lbrakk>x' \<in> S; dist x x' < d\<rbrakk> \<Longrightarrow> (\<forall>y\<in>f x. \<exists>y'. y' \<in> f x' \<and> dist y y' < e) \<and> (\<forall>y'\<in>f x'. \<exists>y. y \<in> f x \<and> dist y' y < e)"
plus_fun_def: fixes m1 :: "'c \<Rightarrow> 'd"   and m2 :: "'c \<Rightarrow> 'd" shows "m1 + m2 \<equiv> \<lambda>x. if m2 x = none then m1 x else m2 x"
is_full: fixes a :: "'a"   and a' :: "'a"   and g :: "'a" assumes "S.ide a"   and "S.ide a'"   and "\<guillemotleft>g : L a' \<Rightarrow>\<^sub>S L a\<guillemotright>" shows "\<exists>f. \<guillemotleft>f : a' \<Rightarrow>\<^sub>S a\<guillemotright> \<and> L f = g"
simps_1: fixes ts :: "'b set set" shows "partitioning_nontail [] ts = ts"
diff2_1: shows "local.rev_curr f m \<noteq> local.rev_curr g m"
contr: assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> Fault f" shows "False"
prob_space_T': fixes K :: "'c \<Rightarrow> 'c pmf"   and I :: "'c pmf" shows "prob_space (MC_syntax.T' K I)"
smc_is_arrD_2: fixes a :: "V"   and b :: "V"   and f :: "V" assumes "f : a \<mapsto>\<^bsub>\<BB>\<^esub> b" shows "a \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>"
body_def: shows "body \<equiv> \<lambda>(C, m). bdy (the (method C m))"
axioms_3: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and u_block_size :: "int"   and grouping :: "int"   and index :: "int" assumes "t_design point_set block_collection u_block_size grouping index" shows "t_design_axioms u_block_size grouping"
le_inf_iff: fixes inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and x :: "'b::type"   and y :: "'b::type"   and z :: "'b::type" assumes "class.semilattice_inf (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool)" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (x::'b::type) ((inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (y::'b::type) (z::'b::type)) = (less_eq x y \<and> less_eq x z)"
mult_sign_intros_5: fixes a :: "'a"   and b :: "'a" assumes "(0::'a) < a"   and "(0::'a) < b" shows "(0::'a) < a * b"
contains_def: fixes A :: "'e set"   and x :: "'e" shows "Predicate_Compile.contains A x = (x \<in> A)"
fmd_optI: fixes P :: "cld list"   and ty :: "ty"   and ctxclds :: "(ctx \<times> cld) list"   and meth :: "char list"   and ctxmeth_def_opt :: "(ctx \<times> meth_def) option" assumes "find_path_ty P ty (Some ctxclds)"   and "find_meth_def_in_path ctxclds meth ctxmeth_def_opt" shows "find_meth_def P ty meth ctxmeth_def_opt"
class_compP: fixes P :: "'a prog"   and C :: "char list"   and D :: "char list"   and fs :: "fdecl list"   and ms :: "'a mdecl list"   and f :: "staticb \<Rightarrow> 'a \<Rightarrow> 'b" assumes "class P C = \<lfloor>(D, fs, ms)\<rfloor>" shows "class (compP f P) C = \<lfloor>(D, fs, map (compM f) ms)\<rfloor>"
support: shows "pmf (sds R26) c = 0 \<or> pmf (sds R26) b = 0"
\<F>_suffix: fixes \<phi> :: "'a ltln"   and i :: "nat"   and w :: "nat \<Rightarrow> 'a set" shows "\<F> \<phi> (suffix i w) \<subseteq> \<F> \<phi> w"
orb_inv_ss: fixes g :: "'a \<Rightarrow> 'a"   and y :: "'a" assumes "permutation g"   and "y \<in> orbit (inv g) x" shows "y \<in> orbit g x"
elims_2: fixes x :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and xa :: "'a list"   and xb :: "'b list"   and xc :: "'c list" assumes "list_all3 x xa xb xc" obtains P :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool" and xd :: "'a" and xs :: "'a list" and y :: "'b" and ys :: "'b list" and z :: "'c" and zs :: "'c list" where "x = P__" and "xa = xd # xs" and "xb = y # ys" and "xc = z # zs" and "P__ xd y z \<and> list_all3 P__ xs ys zs"   | P :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool" where "x = P" and "xa = []" and "xb = []" and "xc = []"
exists_HaddP: fixes H :: "fm set"   and y :: "tm"   and j :: "name"   and x :: "tm" assumes "H \<turnstile> OrdP y"   and "atom j \<sharp> (x, y)" shows "H \<turnstile> SyntaxN.Ex j (HaddP x y (Var j))"
pol_deg_le_n: fixes p :: "'a"   and c :: "nat \<times> (nat \<Rightarrow> 'a)" assumes "p \<in> carrier R"   and "pol_coeff S c"   and "p = polyn_expr R X (fst c) c" shows "deg_n R S X p \<le> fst c"
dist_triangle: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and x :: "'c"   and z :: "'c"   and y :: "'c" assumes "class.metric_space dist uniformity open" shows "dist x z \<le> dist x y + dist y z"
anga_distinct: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.QCongAAcute Bet Cong a"   and "a A B C" shows "A \<noteq> B \<and> C \<noteq> B"
cUN_empty: fixes B :: "'e \<Rightarrow> 'd cset" shows "cUnion (cimage B cempty) = cempty"
freeliftid_kernel: fixes c :: "'w freeword" assumes "c \<in> FreeGroup S"   and "freeword_funlift id c = (0::'w)" shows "c \<in> \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` Abs_freeword ` map (charpair S) ` R\<rangle>"
distinct_28: fixes x41 :: "('a, 'b, 'c) Stmt"   and x42 :: "('a, 'b, 'c) Stmt"   and x31 :: "('a, 'b, 'c) Stmt"   and x32 :: "'a ModeUpd" shows "x41 ;; x42 \<noteq> x31@[x32]"
assms_3: shows "\<forall>c\<in>clk_set A. v c \<le> n"
Pred\<^sub>\<alpha>_eq_iff: fixes \<phi>1 :: "'f"   and \<phi>2 :: "'f" shows "(Pred\<^sub>\<alpha> \<phi>1 = Pred\<^sub>\<alpha> \<phi>2) = (\<phi>1 = \<phi>2)"
curryE: fixes f :: "'c \<times> 'd \<Rightarrow> bool"   and a :: "'c"   and b :: "'d" assumes "curry f a b" obtains "f (a, b)"
Problem_axioms: shows "Problem wtFsym wtPsym arOf resOf parOf \<Phi>"
inv\<^sub>3I: fixes P\<^sub>1 :: "'a set set"   and P\<^sub>2 :: "'a set set"   and B\<^sub>1 :: "'a set"   and B\<^sub>2 :: "'a set"   and S :: "'a set"   and L :: "'a set" assumes "inv\<^sub>2 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 S L"   and "B\<^sub>2 \<subseteq> {u \<in> U. w u \<le> real c / 2}" shows "inv\<^sub>3 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 S L"
fake_monotone_param1: fixes T :: "atom set"   and T' :: "atom set"   and U :: "fid_t set"   and H :: "chmsg set" assumes "T \<subseteq> T'" shows "fake T U H \<subseteq> fake T' U H"
right_inverse: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and inverse :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "group f z inverse" shows "f a (inverse a) = z"
field_vector_diff_chain_within: fixes f :: "real \<Rightarrow> 'a"   and f' :: "'a"   and x :: "real"   and S :: "real set"   and g :: "'a \<Rightarrow> 'a"   and g' :: "'a" assumes "(f has_vector_derivative f') (at x within S)"   and "(g has_field_derivative g') (at (f x) within f ` S)" shows "(g \<circ> f has_vector_derivative f' * g') (at x within S)"
homotopic_loops_sym_eq: fixes s :: "'a set"   and p :: "real \<Rightarrow> 'a"   and q :: "real \<Rightarrow> 'a" shows "homotopic_loops s p q = homotopic_loops s q p"
length_cons_def: fixes g :: "('e, 'f) generator"   and s :: "'f" shows "length_cons g s = length (unstream g s)"
Supq_distl: fixes x :: "'a"   and Y :: "'a set" shows "x \<cdot> \<Sqinter> Y = \<Sqinter> ((\<cdot>) x ` Y)"
select_convs_1: fixes ISignature :: "('a, 'b) Signature"   and ICategory :: "('c, 'd) Category"   and ITypes :: "'a \<Rightarrow> 'c"   and IFunctions :: "'b \<Rightarrow> 'd"   and more :: "'e" shows "iS\<^bsub>\<lparr>ISignature = ISignature, ICategory = ICategory, ITypes = ITypes, IFunctions = IFunctions, \<dots> = more\<rparr>\<^esub> = ISignature"
D_def: shows "D = {x \<in> {0<..(int p * int q - 1) div 2}. x mod int p \<in> {(int p - 1) div 2<..int p - 1} \<and> x mod int q \<in> {(int q - 1) div 2<..int q - 1}}"
sup_mono: fixes a :: "'a"   and c :: "'a"   and b :: "'a"   and d :: "'a" assumes "a \<le> c"   and "b \<le> d" shows "a + b \<le> c + d"
correctCompositionDiffLevelsS8: shows "correctCompositionDiffLevels sS8"
ANR_homeomorphic_ANR: fixes T :: "'a set"   and S :: "'b set" assumes "ANR T"   and "S homeomorphic T" shows "ANR S"
le_nb: fixes t :: "tm" assumes "tmbound0 t" shows "bound0 (le t)"
60: shows "rank A w ` sset (gtrace r v) \<subseteq> {k}"
CallRedsThrowParams: fixes e :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>0 :: "bool"   and v :: "val"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>1 :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and es :: "char list exp list"   and vs\<^sub>1 :: "val list"   and a :: "char list exp"   and es\<^sub>2 :: "char list exp list"   and s\<^sub>2 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>2 :: "bool"   and M :: "char list" assumes "P \<turnstile> \<langle>e,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>Val v,s\<^sub>1,b\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>es,s\<^sub>1,b\<^sub>1\<rangle> [\<rightarrow>]* \<langle>map Val vs\<^sub>1 @ throw a # es\<^sub>2,s\<^sub>2,b\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>e\<bullet>M(es),s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>throw a,s\<^sub>2,b\<^sub>2\<rangle>"
1: shows "d \<squnion> selected_edge h j g \<le> - forest_components h"
D_def: shows "(D::'a::refinement_lattice set) = {d \<parallel> (i::'a::refinement_lattice) |d::'a::refinement_lattice. (c::'a::refinement_lattice) \<sqsubseteq> d \<parallel> i}"
split_llist_first: fixes x :: "'b"   and xs :: "'b llist" assumes "x \<in> lset xs" shows "\<exists>ys zs. xs = ys $ x % zs \<and> lfinite ys \<and> x \<notin> lset ys"
simps_4: fixes a :: "name"   and b :: "name"   and P :: "pi" shows "depth (a{b}.P) = 1 + depth P"
isomorphic_def: fixes a :: "'a"   and a' :: "'a" shows "C.isomorphic a a' = (\<exists>f. \<guillemotleft>f : a \<Rightarrow>\<^sub>C a'\<guillemotright> \<and> C.iso f)"
lhd_conv_lnth: fixes xs :: "'a llist" assumes "\<not> lnull xs" shows "lhd xs = lnth xs 0"
prems_2: fixes l1 :: "'a list"   and l2 :: "'a list" assumes "a # l = l1 @ e # l2" shows "P"
suminf_divide: fixes f :: "nat \<Rightarrow> 'a" assumes "summable f" shows "(\<Sum>n. f n / c) = suminf f / c"
prev: shows "S = UNIV \<or> (\<exists>f g. f holomorphic_on S \<and> g holomorphic_on ball 0 1 \<and> (\<forall>z\<in>S. f z \<in> ball 0 1 \<and> g (f z) = z) \<and> (\<forall>z\<in>ball 0 1. g z \<in> S \<and> f (g z) = z))"
P7: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x l\<rightarrow> y r\<rightarrow> z = (1::'a)) = (y r\<rightarrow> x l\<rightarrow> z = (1::'a))"
lowest_vertex_def: fixes X :: "'a \<times> 'a pdevs" shows "lowest_vertex X = fst X - sum_list (map snd (list_of_pdevs (snd X)))"
closed_halfspace_Im_ge: fixes b :: "real" shows "closed {z. b \<le> Im z}"
periodic_fun_simple_axioms: shows "periodic_fun_simple cos ((2::'b) * of_real pi)"
replacement: fixes x :: "hf"   and R :: "hf \<Rightarrow> hf \<Rightarrow> bool" assumes "\<forall>u v v'. u \<^bold>\<in> x \<longrightarrow> R u v \<longrightarrow> R u v' \<longrightarrow> v' = v" shows "\<exists>z. \<forall>v. (v \<^bold>\<in> z) = (\<exists>u. u \<^bold>\<in> x \<and> R u v)"
ln_fact_conv_sum_upto_mangoldt: fixes n :: "nat" shows "ln (fact n) = sum_upto (\<lambda>k. mangoldt k * real (n div k)) (real n)"
path_connectedin_path_component_of: fixes X :: "'a topology"   and x :: "'a" shows "path_connectedin X (path_component_of_set X x)"
bounded_quasi_semilattice_axioms: shows "bounded_quasi_semilattice gcd (0::'a) (1::'a) normalize"
sums_NSsums_iff: fixes f :: "nat \<Rightarrow> real"   and l :: "real" shows "f sums l = f NSsums l"
Map_ide_2: fixes a :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr" assumes "J_C.ide a" shows "J_C.Map a = J_C.Cod a"
assms_2: shows "B o\<^sub>C\<^sub>L A = id_cblinfun"
antisym: fixes \<delta> :: "'e \<Rightarrow> 'f \<times> 'g \<Rightarrow> 'e"   and \<gamma> :: "'e \<Rightarrow> 'f \<times> 'g \<Rightarrow> 'h"   and bot :: "'e"   and s1 :: "'e"   and s2 :: "'e" assumes "RDR \<delta> \<gamma> bot"   and "pre_RDR.less_eq \<delta> s1 s2 \<and> pre_RDR.less_eq \<delta> s2 s1" shows "s1 = s2"
of_char_eqI: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "char"   and d :: "char" assumes "class.unique_euclidean_semiring_with_bit_shifts divide plus minus zero times one modulo euclidean_size division_segment bit push_bit drop_bit take_bit"   and "comm_semiring_1.of_char times one plus zero c = comm_semiring_1.of_char times one plus zero d" shows "c = d"
simps: 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 cms :: "'a \<times> (Mode \<Rightarrow> 'b set)"   and mem :: "'b \<Rightarrow> 'c"   and cms' :: "'a \<times> (Mode \<Rightarrow> 'b set)"   and mem' :: "'b \<Rightarrow> 'c"   and X :: "'b set"   and g :: "('b \<Rightarrow> 'c option) \<Rightarrow> 'b \<Rightarrow> 'c option" assumes "sifum_security stop eval some_val some_val'" shows "sifum_security.change_respecting eval (cms, mem) (cms', mem') X g = (((cms, mem), cms', mem') \<in> eval \<and> (\<forall>\<sigma>. dom \<sigma> = X \<longrightarrow> g \<sigma> \<preceq> \<sigma>) \<and> (\<forall>\<sigma> \<sigma>'. dom \<sigma> = X \<and> dom \<sigma>' = X \<longrightarrow> dom (g \<sigma>) = dom (g \<sigma>')) \<and> (\<forall>\<sigma>. dom \<sigma> = X \<longrightarrow> ((cms, subst \<sigma> mem), cms', subst (g \<sigma>) mem') \<in> eval))"
f_def: shows "f = (\<lambda>t. if t \<le> dist x y then g t else h t)"
comp_red_basis_is_auto_reduced: fixes xs :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "is_auto_reduced (set (comp_red_basis xs))"
len_faces_makeFaceFinal: fixes f :: "face"   and g :: "graph" shows "|faces (makeFaceFinal f g)| = |faces g|"
aaba: shows "(aa, ba) = mk_v I (OVar vid1) (sol 0, bb) (sol t)"
free_strict_monoidal_category_axioms: shows "free_strict_monoidal_category C"
Minj_pred_def: fixes i :: "nat"   and M :: "mon" shows "Minj_pred i M = (if i = 1 then M else mkMinj (i - 1) M)"
rGamma_complex_real: fixes z :: "complex" assumes "z \<in> \<real>" shows "rGamma z \<in> \<real>"
Inter_in_lattice: fixes M :: "'b set set"   and L :: "'b set set" assumes "M \<subseteq> L"   and "lattice L" shows "\<Inter> M \<in> L"
equivalent_on_def: fixes \<sigma> :: "('b \<times> 'b trm) list"   and \<eta> :: "('b \<times> 'b trm) list"   and V :: "'b set"   and I :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool" shows "equivalent_on \<sigma> \<eta> V I = (\<forall>x\<in>V. I (Var x \<lhd> \<sigma>) (Var x \<lhd> \<eta>))"
that: fixes h :: "real \<Rightarrow> real" assumes "h \<in> o(\<lambda>x. ln (b * x) powr - (e / 2 + 1))"   and "\<forall>\<^sub>F x in at_top. ln (b * x) powr (- e / 2) + h x \<le> f x"   and "\<forall>\<^sub>F x in at_top. \<bar>ln (b * x) powr (- e / 2) + h x\<bar> < 1" shows "thesis"
monic_power: fixes p :: "'a poly"   and n :: "nat" assumes "monic p" shows "monic (p ^ n)"
sum_gp_basic: fixes x :: "'b"   and n :: "nat" shows "((1::'b) - x) * sum ((^) x) {..n} = (1::'b) - x ^ Suc n"
sem_5ab: fixes X :: "w \<Rightarrow> bool"   and Y :: "w \<Rightarrow> bool" shows "ob X Y \<longrightarrow> \<I>X \<sqinter> Y"
measurable_restrict_space2: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and \<Omega> :: "'c set"   and N :: "'c measure" assumes "f \<in> space M \<rightarrow> \<Omega>"   and "f \<in> M \<rightarrow>\<^sub>M N" shows "f \<in> M \<rightarrow>\<^sub>M restrict_space N \<Omega>"
assms: shows "((x0, x1), w) \<in> Rel_OR"
pnormalize_unique: fixes p :: "'a list"   and q :: "'a list" assumes "poly p = poly q" shows "pnormalize p = pnormalize q"
good_ex: fixes G :: "('c, 'd) ParityGame_scheme"   and S :: "'c set"   and p :: "Player"   and good :: "'c \<Rightarrow> ('c \<Rightarrow> 'c) set"   and r :: "(('c \<Rightarrow> 'c) \<times> ('c \<Rightarrow> 'c)) set"   and v :: "'c" assumes "WellOrderedStrategies G S p good r"   and "v \<in> S" shows "\<exists>\<sigma>. \<sigma> \<in> good v"
assms_3: fixes i :: "'a" assumes "i \<in> Basis" shows "continuous_on s (\<lambda>x. f x \<bullet> i)"
mset_subset_of_empty: fixes A :: "'k multiset" shows "(A \<subset># {#}) = False"
vsubstEnv_psubstEnv_compose: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and rho' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and ys :: "'varSort"   and y1 :: "'var"   and y :: "'var" assumes "goodEnv rho"   and "goodEnv rho'" shows "rho &[rho'] &[y1 // y]_ys = rho &[rho' &[y1 // y]_ys]"
nextR_def: shows "nextR = (\<Lambda> r. case ldropWhile\<cdot>(\<Lambda> x. SetMem\<cdot>x\<cdot>(exceptR\<cdot>r))\<cdot>(resultR\<cdot>r) of lnil \<Rightarrow> Nothing | x :@ xs \<Rightarrow> Just\<cdot>(x, R\<cdot>xs\<cdot>(exceptR\<cdot>r)))"
execute_parallel_operators_strips_none_if: fixes ops :: "'b strips_operator list"   and v :: "'b"   and s :: "'b \<Rightarrow> bool option" assumes "\<forall>op\<in>set ops. v \<notin> set (add_effects_of op) \<and> v \<notin> set (delete_effects_of op)"   and "s v = None" shows "execute_parallel_operator s ops v = None"
zero_base_lsb_imp_set_eq_as_bit_operation: fixes len :: "nat"   and base :: "'a word"   and a :: "'a word" assumes "mask (LENGTH('a) - len) && base = 0" shows "(base = ~~ mask (LENGTH('a) - len) && a) = (a \<in> {base..base || mask (LENGTH('a) - len)})"
simps_3: fixes ds :: "bool list"   and T\<^sub>1 :: "tree"   and T\<^sub>2 :: "tree" shows "delete (False # ds) (Branching T\<^sub>1 T\<^sub>2) = Branching T\<^sub>1 (delete ds T\<^sub>2)"
simps_1: fixes P :: "poly \<Rightarrow> bool"   and c :: "poly" shows "allpolys P (CP c) = P c"
map\<^sub>0_simp: fixes a :: "'d" assumes "D.obj a" shows "I\<^sub>D.map\<^sub>0 a = a"
commute: fixes a :: "'c multiset"   and b :: "'c multiset" shows "a \<inter># b = b \<inter># a"
chain_Sync1: fixes Y :: "nat \<Rightarrow> 'b process"   and A :: "'b set"   and S :: "'b process" assumes "chain Y" shows "chain (\<lambda>i. Y i \<lbrakk>A\<rbrakk> S)"
13: shows "r\<^sup>T * (v * v\<^sup>T \<sqinter> g)\<^sup>\<star> * e * (e\<^sup>T * v * v\<^sup>T \<sqinter> g) \<le> r\<^sup>T * (v * v\<^sup>T \<sqinter> g)\<^sup>\<star>"
NOT_mask_shifted_lenword: fixes len :: "nat" shows "~~ (mask len << LENGTH('b) - len) = mask (LENGTH('b) - len)"
allvw: shows "allvw = \<lbrakk>quot_dbfm (abst_dbfm (decode_Var v) 0 (trans_fm [] (SyntaxN.All (decode_Var w) (A IMP Aw IMP Ae))))\<rbrakk>e"
errMOD_igSubstIGVar1STR: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'a, 'g, 'h) model" assumes "FixSyn TYPE('a) Delta"   and "FixSyn.igVarIPresIGWls Delta MOD"   and "FixSyn.igSubstIGVar1 Delta MOD" shows "FixSyn.igSubstIGVar1STR Delta (FixSyn.errMOD Delta MOD)"
acyclic_3d_def: fixes x :: "'a" shows "acyclic_3d x \<equiv> \<forall>y. y \<le> x \<and> y\<^sup>+ = x\<^sup>+ \<longrightarrow> y = x"
dfs_match_subset: fixes f :: "('d, 'e) bintree \<Rightarrow> 'c"   and a :: "'d"   and xs :: "('d, 'e) bintree option list" shows "set (dfs f (map (match a) xs)) \<subseteq> set (dfs f xs)"