hcju/isabelleps huggingface dataset:
refine: fixes E :: "nat"   and slpi :: "floatarith list"   and slp :: "floatarith list"   and Xi :: "'b list"   and X :: "real list set"   and fas :: "floatarith list" assumes "DIM_precond TYPE('c) E"   and "(slpi, slp) \<in> fas_rel"   and "(Xi, X) \<in> appr_rell" shows "(nres_of (approx_slp_appr_impl E slpi Xi), approx_slp_appr fas $ slp $ X) \<in> \<langle>appr_rel\<rangle>nres_rel"
keysFor_Un: fixes H :: "msg set"   and H' :: "msg set" shows "keysFor (H \<union> H') = keysFor H \<union> keysFor H'"
gs1: shows "image_mset pl.Mp (mset gs1) = {#x \<in># image_mset pl.Mp (mset gs). x dvdm f1#}"
ipurge_ref_append: fixes I :: "('d \<times> 'd) set"   and D :: "'c \<Rightarrow> 'd"   and u :: "'d"   and xs :: "'c list"   and ys :: "'c list"   and X :: "'c set" shows "ipurge_ref I D u (xs @ ys) X = ipurge_ref_aux I D (insert u (sinks I D u xs)) ys X"
invfun_r1: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'f set"   and b :: "'f" assumes "f \<in> A \<rightarrow> B"   and "bij_to f A B"   and "b \<in> B" shows "f ((f\<inverse>\<^bsub>B,A\<^esub>) b) = b"
Abs_Sequence\<^sub>b\<^sub>a\<^sub>s\<^sub>e_inverse: fixes y :: "\<langle>\<langle>'d list\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "y \<in> {X. X = \<bottom> \<or> X = null \<or> (\<forall>x\<in>set \<lceil>\<lceil>X\<rceil>\<rceil>. x \<noteq> \<bottom>)}" shows "Rep_Sequence\<^sub>b\<^sub>a\<^sub>s\<^sub>e (Abs_Sequence\<^sub>b\<^sub>a\<^sub>s\<^sub>e y) = y"
const_type_inv: fixes arity :: "'a \<Rightarrow> nat"   and public :: "'a \<Rightarrow> bool"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and \<Gamma> :: "('a, 'b) Term.term \<Rightarrow> ('a, 'c) Term.term"   and c :: "'a"   and X :: "('a, 'b) Term.term list"   and a :: "'c" assumes "typed_model arity public Ana \<Gamma>"   and "\<Gamma> (Fun c X) = Var a" shows "arity c = 0"
finite_imp_nat_seg_image_inj_on: fixes A :: "'b set" assumes "finite A" shows "\<exists>n f. A = f ` {i. i < n} \<and> inj_on f {i. i < n}"
Mf': shows "(\<lambda>z. \<integral>\<^sup>+ x. f (x, z - x) \<partial>count_space UNIV) \<in> borel_measurable (count_space UNIV)"
ah_leq_il_right: fixes h1 :: "'a \<Rightarrow> 'a set"   and h2 :: "'a \<Rightarrow> 'a set"   and h2' :: "'a \<Rightarrow> 'a set" assumes "h1 [*] h2"   and "h2' \<le> h2" shows "h1 [*] h2'"
hyps_4: fixes b :: "trm" assumes "\<Gamma> \<turnstile> t \<Leftrightarrow> b : T\<^sub>1" shows "\<Gamma> \<turnstile> s \<Leftrightarrow> b : T\<^sub>1"
sym_mpoly_recurrence: fixes k :: "nat"   and X :: "nat set" assumes "0 < k"   and "finite X" shows "sym_mpoly X k = - MPoly_Type.smult ((1::'a) / of_nat k) (\<Sum>i = 1..k. (- 1) ^ i * sym_mpoly X (k - i) * powsum_mpoly X i)"
set_child_nodes_get_child_nodes: fixes ptr :: "(_) object_ptr"   and h :: "(_) heap"   and children :: "(_) node_ptr list"   and h' :: "(_) heap" assumes "ShadowRootClass.known_ptr ptr"   and "ShadowRootClass.type_wf h"   and "h \<turnstile> Shadow_DOM.set_child_nodes ptr children \<rightarrow>\<^sub>h h'" shows "h' \<turnstile> Shadow_DOM.get_child_nodes ptr \<rightarrow>\<^sub>r children"
th': fixes x :: "nat"   and y :: "nat" shows "\<not> (x \<noteq> y \<and> \<not> x < y \<and> (\<forall>m. x \<noteq> y + m))"
eexp_special_values_2: shows "eexp \<infinity> = \<infinity>"
add_strict_mono: fixes a :: "'c multiset"   and b :: "'c multiset"   and c :: "'c multiset"   and d :: "'c multiset" assumes "a \<subset># b"   and "c \<subset># d" shows "a + c \<subset># b + d"
SD_def: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and q :: "'b pmf"   and p :: "'b pmf" shows "SD R q p = (p \<in> lotteries_on {x. R x x} \<and> q \<in> lotteries_on {x. R x x} \<and> (\<forall>x. R x x \<longrightarrow> measure_pmf.prob q {y. R x y} \<le> measure_pmf.prob p {y. R x y}))"
someI: fixes P :: "'g \<Rightarrow> bool"   and x :: "'g" assumes "P x" shows "P (Eps P)"
assms: shows "nP = (\<lambda>v. case BLP_security_levels v of None \<Rightarrow> SINVAR_BLPtrusted.default_node_properties | Some c \<Rightarrow> c)"
P1: shows "C B J' CongA C B J \<and> \<not> B C OS A J' \<and> \<not> A B TS C J' \<and> Coplanar A B C J'"
append_insert: fixes xs :: "'b list"   and x :: "'b" shows "set (xs @ [x]) = insert x (set xs)"
lt_max: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and v :: "'a" assumes "lookup p v \<noteq> (0::'c)" shows "v \<preceq> punit.lt p"
Urysohn_local_strong: fixes U :: "'c set"   and S :: "'c set"   and T :: "'c set"   and a :: "'d"   and b :: "'d" assumes "closedin (top_of_set U) S"   and "closedin (top_of_set U) T"   and "S \<inter> T = {}"   and "a \<noteq> b" obtains f :: "'c \<Rightarrow> 'd" where "continuous_on U f" and "\<And>x. x \<in> U \<Longrightarrow> f x \<in> closed_segment a b" and "\<And>x. x \<in> U \<Longrightarrow> (f x = a) = (x \<in> S)" and "\<And>x. x \<in> U \<Longrightarrow> (f x = b) = (x \<in> T)"
cobounded_imp_unbounded: fixes S :: "'b set" assumes "bounded (- S)" shows "\<not> bounded S"
prob_eq_0: fixes M :: "'b measure"   and A :: "'b set" assumes "prob_space M"   and "A \<in> sets M" shows "(Sigma_Algebra.measure M A = 0) = (AE x in M. x \<notin> A)"
diff_eq1: fixes x :: "'a"   and y :: "'a" assumes "x \<in> E"   and "y \<in> E" shows "x - y = x + - y"
hom_ab: shows "homeomorphism (cbox a b) (cbox u v) f1 g1"
simps_5: fixes f1 :: "nat \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and x1 :: "nat" shows "(case Prime x1 of Prime x \<Rightarrow> f1 x | Triple x xa xb \<Rightarrow> f2 x xa xb) = f1 x1"
Cons: fixes y :: "'b"   and ys' :: "'b list" assumes "ys = y # ys'"   and "R x y"   and "list_all2 R xs' ys'" shows "phi"
simps_2: fixes x :: "'c"   and xs :: "'c list"   and ys :: "'d list" shows "List.product (x # xs) ys = map (Pair x) ys @ List.product xs ys"
asymptotics: shows "(\<lambda>x. sum_upto (\<lambda>n. ind prime n * ln (real n) / real n) x - ln x) \<in> O(\<lambda>_. 1)"
ide_pbdom: fixes f :: "'a"   and g :: "'a" assumes "C.cospan f g" shows "C.ide (f \<down>\<down> g)"
absorb2: fixes b :: "'b"   and a :: "'b" assumes "b \<le> a" shows "a \<sqinter> b = b"
DirProds_m_closed: fixes x :: "'e \<Rightarrow> 'f"   and G :: "'e \<Rightarrow> ('f, 'g) monoid_scheme"   and I :: "'e set"   and y :: "'e \<Rightarrow> 'f" assumes "x \<in> carrier (DirProds G I)"   and "y \<in> carrier (DirProds G I)"   and "\<And>i. i \<in> I \<Longrightarrow> Group.group (G i)" shows "x \<otimes>\<^bsub>DirProds G I\<^esub> y \<in> carrier (DirProds G I)"
mult_sub_right_one: fixes x :: "'a" shows "x \<le> x \<sqinter> top"
strictly_precedes_sub: fixes sub :: "'b run"   and r :: "'b run"   and c\<^sub>1 :: "clock"   and c\<^sub>2 :: "clock" assumes "sub \<lless> r"   and "sub \<in> \<lbrakk> c\<^sub>1 strictly precedes c\<^sub>2 \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L" shows "r \<in> \<lbrakk> c\<^sub>1 strictly precedes c\<^sub>2 \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L"
subset_Un_eq: fixes A :: "'d set"   and B :: "'d set" shows "(A \<subseteq> B) = (A \<union> B = B)"
fun_add_closed: fixes f :: "(nat \<Rightarrow> int) \<Rightarrow> nat \<Rightarrow> int"   and g :: "(nat \<Rightarrow> int) \<Rightarrow> nat \<Rightarrow> int" assumes "f \<in> carrier (function_ring (carrier Zp) Zp)"   and "g \<in> carrier (function_ring (carrier Zp) Zp)" shows "f \<oplus>\<^bsub>function_ring (carrier Zp) Zp\<^esub> g \<in> carrier (function_ring (carrier Zp) Zp)"
iterate_size_def: fixes it :: "(nat \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> nat \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat" shows "iterate_size it = it (\<lambda>_. True) (\<lambda>x. Suc) 0"
ipset_from_cidr_alt: fixes pre :: "'a word"   and len :: "nat" shows "ipset_from_cidr pre len = {pre && (mask len << LENGTH('a) - len)..pre || mask (LENGTH('a) - len)}"
suffix_drop: fixes n :: "nat"   and as :: "'a list" shows "suffix (drop n as) as"
part_overlap_eq: fixes x :: "i"   and y :: "i" shows "P x y = (\<forall>z. O z x \<longrightarrow> O z y)"
entails_frame': fixes A :: "assn"   and F :: "assn"   and Q :: "assn"   and B :: "assn" assumes "\<not> (A * F \<Longrightarrow>\<^sub>A Q)"   and "A \<Longrightarrow>\<^sub>A B" shows "\<not> (B * F \<Longrightarrow>\<^sub>A Q)"
assms_3: shows "f'' : a'' \<mapsto>\<^bsub>\<CC>\<^esub> b''"
wf_chain_fst: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and r :: "'c rule"   and rs :: "'c rule list" assumes "wf_chain \<Gamma> (r # rs)" shows "wf_chain \<Gamma> rs"
subst_freshen_assumptions: fixes pf :: "'form"   and s :: "'subst"   and a :: "nat" assumes "pf \<in> set assumptions" shows "subst s (freshen a pf) = pf"
l11_16: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p" assumes "Per A B C"   and "A \<noteq> B"   and "C \<noteq> B"   and "Per A' B' C'"   and "A' \<noteq> B'"   and "C' \<noteq> B'" shows "A B C CongA A' B' C'"
Ide_in_Hom: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b monoidal_language.term" assumes "monoidal_language C"   and "monoidal_language.Ide C t" shows "t \<in> {t. monoidal_language.Arr C t \<and> monoidal_language.Dom C t = t \<and> monoidal_language.Cod C t = t}"
less_eq_inf: fixes x :: "'a"   and y :: "'a" shows "(- x \<le> - y) = (- x \<sqinter> - y = - x)"
lunstream_lstream: fixes xs :: "'c llist" shows "lunstream lstream xs = xs"
terminates_dropWhile_raw: fixes g :: "'e \<Rightarrow> ('f, 'e) step"   and P :: "'f \<Rightarrow> bool" assumes "terminates g" shows "terminates (dropWhile_raw P g)"
g: shows "g \<in> M \<rightarrow>\<^sub>M M1"
compseriesTr0: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a set"   and i :: "nat" assumes "compseries G n f"   and "i \<le> n" shows "G \<guillemotright> f i "
abs_diff_triangle_ineq: fixes abs :: "'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 less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "less_eq (abs (minus (plus a b) (plus c d))) (plus (abs (minus a c)) (abs (minus b d)))"
compare_expansions_def: fixes F :: "'a"   and G :: "'a" shows "compare_expansions F G = (case compare_lists (snd (dominant_term F)) (snd (dominant_term G)) of LT \<Rightarrow> (LT, 0, 0) | EQ \<Rightarrow> (EQ, fst (dominant_term F), fst (dominant_term G)) | GT \<Rightarrow> (GT, 0, 0))"
remove_child_removes_first_child: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and node_ptr :: "(_) node_ptr"   and children :: "(_) node_ptr list"   and h' :: "(_) heap" assumes "Shadow_DOM.heap_is_wellformed h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "h \<turnstile> Shadow_DOM.get_child_nodes ptr \<rightarrow>\<^sub>r node_ptr # children"   and "h \<turnstile> Shadow_DOM.remove_child ptr node_ptr \<rightarrow>\<^sub>h h'" shows "h' \<turnstile> Shadow_DOM.get_child_nodes ptr \<rightarrow>\<^sub>r children"
select_def: fixes k :: "nat"   and xs :: "'a list" shows "select k xs = sort xs ! k"
free_name_simps_2: 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 t :: "'c" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree"   and "pre_term.is_free unfree t" shows "free (pre_term.free_name unfree t) = t"
assoc_mult_mat: fixes A :: "'b mat"   and n\<^sub>1 :: "nat"   and n\<^sub>2 :: "nat"   and B :: "'b mat"   and n\<^sub>3 :: "nat"   and C :: "'b mat"   and n\<^sub>4 :: "nat" assumes "A \<in> carrier_mat n\<^sub>1 n\<^sub>2"   and "B \<in> carrier_mat n\<^sub>2 n\<^sub>3"   and "C \<in> carrier_mat n\<^sub>3 n\<^sub>4" shows "A * B * C = A * (B * C)"
cp_def: shows "cp TYPE('a) TYPE('b) TYPE('c) \<equiv> \<forall>pi2 pi1 x. pi1 \<bullet> pi2 \<bullet> x = (pi1 \<bullet> pi2) \<bullet> pi1 \<bullet> x"
Rep_frame_cases: fixes y :: "'d frame_Rep" assumes "y \<in> Collect frame_Rep_set" obtains x :: "'d frame" where "y = Rep_frame x"
continuous_map_open: fixes T1 :: "'c topology"   and T2 :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and U :: "'d set" assumes "continuous_map T1 T2 f"   and "openin T2 U" shows "openin T1 (f -` U \<inter> topspace T1)"
bv_leqs_antisym: fixes xs :: "bool list"   and ys :: "bool list" assumes "xs \<preceq>\<^sub>b ys"   and "ys \<preceq>\<^sub>b xs" shows "xs = ys"
conv_tail1_3: fixes s :: "'a"   and d :: "bool"   and \<alpha> :: "'b"   and t :: "'a"   and xs :: "(bool \<times> 'b \<times> 'a) list"   and ars :: "('a \<times> 'b \<times> 'a) set" assumes "(s, (d, \<alpha>, t) # xs) \<in> conv ars"   and "\<not> d" shows "(t, \<alpha>, s) \<in> ars"
assms_2: shows "v \<in> dom (\<Pi>\<^sub>I)"
span_finiteE: fixes B :: "'b set"   and p :: "'b" assumes "finite B"   and "p \<in> local.span B" obtains q :: "'b \<Rightarrow> 'a" where "p = (\<Sum>b\<in>B. q b *s b)"
offs_pred_asc_aux: fixes ns :: "nat list"   and ub :: "nat"   and xs :: "'c list"   and index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and mi :: "'d"   and ma :: "'d"   and i :: "nat"   and k :: "nat"   and j :: "nat" assumes "offs_pred ns ub xs index key mi ma"   and "i < length ns"   and "k < length ns"   and "i \<le> j"   and "j < k"   and "0 < offs_num (length ns) xs index key mi ma j"   and "0 < offs_num (length ns) xs index key mi ma k" shows "ns ! j + offs_num (length ns) xs index key mi ma j \<le> ns ! k"
4: shows "\<forall>n. norm (f n * x ^ n) \<le> K"
total_on_less_than: fixes A :: "nat set" shows "total_on A less_than"
mask_2pm1: fixes n :: "nat" shows "mask n = 2 ^ n - 1"
absorb_iff2: fixes a :: "'b"   and b :: "'b" shows "(a \<le> b) = (a \<sqinter> b = b)"
consts_app: fixes size :: "'a \<Rightarrow> nat"   and frees :: "'a \<Rightarrow> name fset"   and subst :: "'a \<Rightarrow> (name, 'a) fmap \<Rightarrow> 'a"   and "consts" :: "'a \<Rightarrow> name fset"   and app :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unapp :: "'a \<Rightarrow> ('a \<times> 'a) option"   and const :: "name \<Rightarrow> 'a"   and unconst :: "'a \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'a"   and unfree :: "'a \<Rightarrow> name option"   and u\<^sub>1 :: "'a"   and u\<^sub>2 :: "'a" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "consts (app u\<^sub>1 u\<^sub>2) = consts u\<^sub>1 |\<union>| consts u\<^sub>2"
Nabla1_subsetof_E1: shows "\<nabla>\<^bsub>\<Gamma>1\<^esub> \<subseteq> E\<^bsub>ES1\<^esub>"
mann10: fixes y :: "'a"   and x :: "'a" shows "y = - (- (- (- 3 \<times> x \<squnion> x) \<squnion> - 3 \<times> x \<squnion> y) \<squnion> - (x \<squnion> y))"
\<dd>\<^sub>P_in_cfl: fixes \<beta> :: "('t + 'n) list"   and \<alpha> :: "('t + 'n) list"   and a :: "'t"   and w :: "'t list" assumes "\<beta> |\<in>| local.\<dd>\<^sub>P \<alpha> a"   and "local.in_cfl \<beta> w" shows "local.in_cfl \<alpha> (a # w)"
freess_cons: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and x :: "'b"   and xs :: "'b list" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "pre_term.freess frees (x # xs) = frees x |\<union>| pre_term.freess frees xs"
zlcms0_iff: fixes "is" :: "int list" shows "(zlcms is = 0) = (0 \<in> set is)"
SN_intro: fixes t :: "trm" assumes "\<And>t'. t \<mapsto> t' \<Longrightarrow> SN t'" shows "SN t"
1: shows "\<turnstile> concl (Thm (Imp (Imp p (Imp (Imp p p) p)) (Imp (Imp p (Imp p p)) (Imp p p))))"
fset_to_im: fixes f :: "'c \<Rightarrow> 'b"   and X :: "'c set" shows "{f x |x. x \<in> X} = f ` X"
Morg: shows "g \<in> mor\<^bsub>C\<^esub>"
phi_bot: fixes phi :: "'c \<Rightarrow> 'd Filters.filter" assumes "triple phi" shows "phi bot = Filters.filter.Abs_filter {top}"
simps_2: fixes \<alpha> :: "'d \<Rightarrow> 'c"   and mc :: "'d monom \<times> 'c"   and p :: "('d monom \<times> 'c) list" shows "eval_poly \<alpha> (mc # p) = eval_monomc \<alpha> mc + eval_poly \<alpha> p"
add_cancel_left_left: fixes plus :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and minus :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero :: "'f"   and b :: "'f"   and a :: "'f" assumes "class.cancel_comm_monoid_add plus minus zero" shows "(plus b a = a) = (b = zero)"
s_Inj1: fixes \<pi> :: "Syntax.term list"   and e :: "Syntax.term"   and m :: "mode"   and \<pi>' :: "Syntax.term list"   and e' :: "Syntax.term" assumes "\<lless>\<pi>, e\<ggreater> m\<rightarrow> \<lless>\<pi>', e'\<ggreater>" shows "\<lless>\<pi>, Inj1 e\<ggreater> m\<rightarrow> \<lless>\<pi>', Inj1 e'\<ggreater>"
iso_unit: fixes a :: "'a" assumes "C.obj a" shows "C.iso \<i>\<^sub>C[a]"
max_mapping_cobounded1: fixes P1 :: "'a \<Rightarrow> 'b option"   and P2 :: "'a \<Rightarrow> 'b option" assumes "dom P1 \<subseteq> dom P2" shows "rel_mapping (\<le>) P1 (max_mapping P1 P2)"
step_member: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and l :: "String.literal"   and i :: "value list"   and t :: "transition"   and s' :: "nat"   and p :: "value option list"   and r' :: "nat \<Rightarrow>f value option" assumes "step e s r l i = Some (t, s', p, r')" shows "(s', t) |\<in>| possible_steps e s r l i"
join_eq: fixes n :: "nat"   and R :: "nat set"   and t1 :: "'b option list set"   and t2 :: "'b option list set"   and pos :: "bool" assumes "table n R t1"   and "table n R t2" shows "join t2 pos t1 = (if pos then t2 \<inter> t1 else t2 - t1)"
lemma_2_5_14: fixes b :: "'a"   and c :: "'a"   and a :: "'a" shows "(b l\<rightarrow> c) * (a l\<rightarrow> b) \<le> a l\<rightarrow> c"
redundant_def: fixes \<alpha>n :: "'a \<Rightarrow> 'b list"   and inEdges' :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) list"   and defs :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and "uses" :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and phis :: "'a \<Rightarrow> 'b \<times> 'd \<Rightarrow> 'd list option"   and g :: "'a" shows "CFG_SSA_wf_base.redundant \<alpha>n inEdges' defs uses phis g \<equiv> \<exists>v\<in>CFG_SSA_base.allVars \<alpha>n inEdges' defs uses phis g. CFG_SSA_wf_base.trivial \<alpha>n inEdges' defs uses phis g v"
pred_stateI: fixes m :: "('c, 'b) state"   and P :: "'b \<Rightarrow> bool" assumes "\<And>a s s'. run_state m s = (a, s') \<Longrightarrow> P a" shows "pred_state P m"
Not\<^sub>\<alpha>_eq_iff: fixes t1\<^sub>\<alpha> :: "('a, 'b, 'c, 'd) Tree\<^sub>\<alpha>"   and t2\<^sub>\<alpha> :: "('a, 'b, 'c, 'd) Tree\<^sub>\<alpha>" shows "(Not\<^sub>\<alpha> t1\<^sub>\<alpha> = Not\<^sub>\<alpha> t2\<^sub>\<alpha>) = (t1\<^sub>\<alpha> = t2\<^sub>\<alpha>)"
simps: fixes a1 :: "db"   and a2 :: "db" shows "(a1 \<Down> a2) = ((\<exists>P M Q V. a1 = DBApp P Q \<and> a2 = V \<and> P \<Down> DBAbsN M \<and> M<Q/0> \<Down> V) \<or> (\<exists>P M Q q V. a1 = DBApp P Q \<and> a2 = V \<and> P \<Down> DBAbsV M \<and> Q \<Down> q \<and> M<q/0> \<Down> V) \<or> (\<exists>e. a1 = DBAbsN e \<and> a2 = DBAbsN e \<and> val (DBAbsN e)) \<or> (\<exists>e. a1 = DBAbsV e \<and> a2 = DBAbsV e \<and> val (DBAbsV e)) \<or> (\<exists>P V. a1 = DBFix P \<and> a2 = V \<and> P<DBFix P/0> \<Down> V) \<or> a1 = DBtt \<and> a2 = DBtt \<or> a1 = DBff \<and> a2 = DBff \<or> (\<exists>C T V E. a1 = DBCond C T E \<and> a2 = V \<and> C \<Down> DBtt \<and> T \<Down> V) \<or> (\<exists>C E V T. a1 = DBCond C T E \<and> a2 = V \<and> C \<Down> DBff \<and> E \<Down> V) \<or> (\<exists>n. a1 = DBNum n \<and> a2 = DBNum n) \<or> (\<exists>P n. a1 = DBSucc P \<and> a2 = DBNum (Suc n) \<and> P \<Down> DBNum n) \<or> (\<exists>P n. a1 = DBPred P \<and> a2 = DBNum n \<and> P \<Down> DBNum (Suc n)) \<or> (\<exists>E. a1 = DBIsZero E \<and> a2 = DBtt \<and> E \<Down> DBNum 0) \<or> (\<exists>E n. a1 = DBIsZero E \<and> a2 = DBff \<and> E \<Down> DBNum n \<and> 0 < n))"
\<alpha>_test: fixes A :: "'s"   and P :: "natural \<Rightarrow> bool" assumes "invar A" shows "local.test A P = Benchmark_Comparison.test (\<alpha> A) P"
simps_6: fixes \<xi> :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltlc"   and \<psi> :: "'b ltlc" shows "\<xi> \<Turnstile>\<^sub>c \<phi> or\<^sub>c \<psi> = (\<xi> \<Turnstile>\<^sub>c \<phi> \<or> \<xi> \<Turnstile>\<^sub>c \<psi>)"
oracle_decrypt_def: fixes k :: "'prf_key \<times> 'upf_key"   and L :: "(bool list \<times> bool list \<times> 'hash) set"   and c :: "bool list \<times> bool list \<times> 'hash" shows "local.ind_cca.oracle_decrypt k L c = return_spmf (if c \<in> L then None else decrypt k c, L)"
\<phi>L: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and num :: "'b set"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and prv :: "'c \<Rightarrow> bool"   and bprv :: "'c \<Rightarrow> bool"   and enc :: "'c \<Rightarrow> 'b"   and S :: "'c"   and P :: "'c"   and \<phi> :: "'c" assumes "Loeb_Form var trm fmla Var num FvarsT substT Fvars subst eql cnj imp all exi prv bprv enc S P"   and "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "Loeb_Form.\<phi>L var fmla Fvars subst cnj imp bprv enc P \<phi> \<in> fmla"
R1_def: shows "R1 = {(S1, S2). \<exists>c1 c1' c2 c2' b b' W W'. S1 = (c1;while b do c2 od) # W \<and> S2 = (c1';while b' do c2' od) # W' \<and> [c1] \<approx>\<^bsub>d\<^esub> [c1'] \<and> [c2] \<approx>\<^bsub>d\<^esub> [c2'] \<and> W \<approx>\<^bsub>d\<^esub> W' \<and> b \<equiv>\<^bsub>d\<^esub> b'}"
run_definitions_2: fixes s\<^sub>0 :: "'b"   and lstep :: "'d \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "nat \<Rightarrow> 'b" assumes "df_system s\<^sub>0 lstep" shows "df_system.is_run s\<^sub>0 lstep s \<equiv> \<exists>l. df_system.is_lrun s\<^sub>0 lstep l s"
map_decr_upt: fixes m :: "nat"   and n :: "nat" shows "map (\<lambda>n. n - Suc 0) [Suc m..<Suc n] = [m..<n]"
allActT: fixes F :: "'c \<Rightarrow> (nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'd" shows "\<turnstile> (\<forall>x. \<box>[F x]_v) = \<box>[(\<forall>x. F x)]_v"
assms_8: shows "is_WAIT (tts (s i) t')"
eq_Nil_appendI: fixes xs :: "'b list"   and ys :: "'b list" assumes "xs = ys" shows "xs = [] @ ys"
cadd_succ: fixes m :: "V"   and n :: "V" shows "ZFC_in_HOL.succ m \<oplus> n = vcard (ZFC_in_HOL.succ (m \<oplus> n))"
simps_3: fixes x :: "nat" shows "fib (Suc (Suc x)) = fib x + fib (Suc x)"
borel_measurable_prod: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and M :: "'d measure" assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M" shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M"
cases: fixes a :: "('i \<Rightarrow> 'a) set" assumes "a \<in> generator" obtains J :: "'i set" and X :: "('i \<Rightarrow> 'a) set" where "a = emb I J X" and "finite J" and "J \<subseteq> I" and "X \<in> sets (Pi\<^sub>M J M)"