hcju/isabelleps huggingface dataset:
post_2: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<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 dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and post :: "'b \<Rightarrow> 'b"   and y :: "'b"   and x :: "'b" assumes "class.post_mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top post" shows "less_eq (inf (times (times y x) top) (post x)) y"
one_pratt2_def: shows "1 \<equiv> [0]\<^sub>2"
x1_l1_props: shows "distinct (x1 # l1) \<and> sorted (x1 # l1)"
order2: fixes p :: "'a list"   and a :: "'a" assumes "poly p \<noteq> poly []" shows "[- a, 1::'a] %^ order a p divides p \<and> \<not> [- a, 1::'a] %^ Suc (order a p) divides p"
atLeastAtMost_subseteq_atLeastLessThan_iff: fixes a :: "'a multiset"   and b :: "'a multiset"   and c :: "'a multiset"   and d :: "'a multiset" shows "(subset_mset.atLeastAtMost a b \<subseteq> subset_mset.atLeastLessThan c d) = (a \<subseteq># b \<longrightarrow> c \<subseteq># a \<and> b \<subset># d)"
matrix_equation_helper_step: fixes p :: "real poly"   and signs :: "rat list list"   and qs :: "real poly list"   and I :: "nat list" assumes "p \<noteq> 0"   and "distinct signs"   and "set (characterize_consistent_signs_at_roots_copr p qs) \<subseteq> set signs"   and "list_constr I (length qs)"   and "\<forall>q. List.member qs q \<longrightarrow> coprime p q" shows "vec_of_list (mtx_row signs I) \<bullet> w_vec p qs signs = rat_of_int (int (card {x. poly p x = 0 \<and> 0 < poly (prod_list (retrieve_polys qs I)) x})) - rat_of_int (int (card {x. poly p x = 0 \<and> poly (prod_list (retrieve_polys qs I)) x < 0}))"
add: fixes b1 :: "'a"   and b2 :: "'a" shows "inner (b1 + b2) (1::'a) = inner b1 (1::'a) + inner b2 (1::'a)"
comp_wb_lens: fixes x :: "'d \<Longrightarrow> 'e"   and y :: "'e \<Longrightarrow> 'f" assumes "wb_lens x"   and "wb_lens y" shows "wb_lens (x ;\<^sub>L y)"
not_N: shows "AE x in lborel. x \<notin> N"
show_vec_real_iarrays_def: fixes v :: "real iarray" shows "show_vec_real_iarrays v = IArray.of_fun (\<lambda>i. show_real (v !! i)) (IArray.length v)"
lspace_approximate_continuous: fixes f :: "'a \<Rightarrow> real"   and p :: "real"   and S :: "'a set"   and e :: "real" assumes "f \<in> lspace (lebesgue_on S) (ennreal p)"   and "S \<in> lmeasurable"   and "1 \<le> p"   and "0 < e" obtains g :: "'a \<Rightarrow> real" where "continuous_on UNIV g" and "g \<in> lspace (lebesgue_on S) (ennreal p)" and "lnorm (lebesgue_on S) p (f - g) < e"
push_precond_eq_sat_or_nonsat: fixes f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and e :: "nat \<times> nat" shows "push_precond f l e = (sat_push_precond f l e \<or> nonsat_push_precond f l e)"
simps_2: fixes uv :: "'d" shows "list_op_deps (Insert None uv) = {}"
DiffI: fixes c :: "'g"   and A :: "'g set"   and B :: "'g set" assumes "c \<in> A"   and "c \<notin> B" shows "c \<in> A - B"
equiintegrable_on_def: fixes F :: "('a \<Rightarrow> 'b) set"   and I :: "'a set" shows "F equiintegrable_on I \<equiv> (\<forall>f\<in>F. f integrable_on I) \<and> (\<forall>e>0. \<exists>\<gamma>. gauge \<gamma> \<and> (\<forall>f \<D>. f \<in> F \<and> \<D> tagged_division_of I \<and> \<gamma> fine \<D> \<longrightarrow> norm ((\<Sum>(x, K)\<in>\<D>. content K *\<^sub>R f x) - integral I f) < e))"
out_out_one_side: fixes A :: "'p"   and B :: "'p"   and X :: "'p"   and Y :: "'p"   and Z :: "'p" assumes "A B OS X Y"   and "A Out Y Z" shows "A B OS X Z"
\<a>_def: fixes \<mu> :: "'a"   and \<nu> :: "'a"   and \<tau> :: "'a" shows "\<a>[\<mu>, \<nu>, \<tau>] \<equiv> B.\<alpha>' (DN (\<mu>, \<nu>, \<tau>))"
6: shows "antipar f (mkarr (Inv (DOM f\<^bold>\<down>) \<^bold>\<cdot> \<^bold>\<lfloor>rep f\<^bold>\<rfloor> \<^bold>\<cdot> COD f\<^bold>\<down>))"
vw'u: shows "(v, w', u) \<in> edges (del_unEdge v w u G)"
image_set: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "f ` set xs = set (list.map f xs)"
assms_1: shows "\<FF> : \<CC> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> dg_prod I \<AA>"
Arr_comp: fixes Obj :: "'a set"   and Hom :: "'a \<Rightarrow> 'a \<Rightarrow> 'b set"   and Id :: "'a \<Rightarrow> 'b"   and Comp :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "('a, 'b) concrete_category.arr"   and g :: "('a, 'b) concrete_category.arr" assumes "concrete_category Obj Hom Id Comp"   and "partial_magma.arr (concrete_category.COMP Obj Hom Comp) f"   and "partial_magma.arr (concrete_category.COMP Obj Hom Comp) g"   and "concrete_category.Dom g = concrete_category.Cod f" shows "concrete_category.COMP Obj Hom Comp g f \<noteq> concrete_category.Null \<and> concrete_category.Dom (concrete_category.COMP Obj Hom Comp g f) \<in> Obj \<and> concrete_category.Cod (concrete_category.COMP Obj Hom Comp g f) \<in> Obj \<and> concrete_category.Map (concrete_category.COMP Obj Hom Comp g f) \<in> Hom (concrete_category.Dom (concrete_category.COMP Obj Hom Comp g f)) (concrete_category.Cod (concrete_category.COMP Obj Hom Comp g f))"
Cd_greatest: fixes d :: "'b"   and X :: "'a set" shows "ManiPref.Cd d X = greatest (override_on Pd Pds' ds' d) (X \<inter> Field (override_on Pd Pds' ds' d))"
ennreal_archimedean: fixes x :: "ennreal" assumes "x \<noteq> \<infinity>" shows "\<exists>xa. x \<le> of_nat xa"
cross3_simps_8: fixes a :: "'b"   and b :: "'b" shows "a + b = b + a"
list_all2_nthD: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'c list"   and p :: "nat" assumes "list_all2 P xs ys"   and "p < length xs" shows "P (xs ! p) (ys ! p)"
Fract_cases: fixes q :: "'b fract" obtains a :: "'b" and b :: "'b" where "q = Fraction_Field.Fract a b" and "b \<noteq> (0::'b)"
abs_eq: shows "(\<le>) = (\<lambda>X Y. Rep_downset X \<subseteq> Rep_downset Y)"
assms_3: shows "x \<in>\<^sub>\<circ> a"
types_preserved_grow_memory: fixes \<S> :: "s_context"   and \<C> :: "t_context"   and c :: "i32"   and ts :: "t list"   and ts' :: "t list"   and c' :: "i32" assumes "\<S>\<bullet>\<C> \<turnstile> [$C ConstInt32 c, $Grow_memory] : ts _> ts'" shows "\<S>\<bullet>\<C> \<turnstile> [$C ConstInt32 c'] : ts _> ts'"
ringModule_def: fixes R :: "('h, 'i) Ring_scheme" shows "M\<^bsub>R\<^esub> = \<lparr>carrier = carrier R, pop = (\<plusminus>\<^bsub>R\<^esub>), mop = mop R, zero = \<zero>\<^bsub>R\<^esub>, sprod = (\<cdot>\<^sub>r\<^bsub>R\<^esub>)\<rparr>"
tranclp_induct3: fixes r :: "'a \<times> 'b \<times> 'c \<Rightarrow> 'a \<times> 'b \<times> 'c \<Rightarrow> bool"   and ax :: "'a"   and ay :: "'b"   and az :: "'c"   and bx :: "'a"   and "by" :: "'b"   and bz :: "'c"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "r\<^sup>+\<^sup>+ (ax, ay, az) (bx, by, bz)"   and "\<And>a aa b. r (ax, ay, az) (a, aa, b) \<Longrightarrow> P a aa b"   and "\<And>a aa b ab ac ba. \<lbrakk>r\<^sup>+\<^sup>+ (ax, ay, az) (a, aa, b); r (a, aa, b) (ab, ac, ba); P a aa b\<rbrakk> \<Longrightarrow> P ab ac ba" shows "P bx by bz"
sub_set_system_axioms: shows "sub_set_system \<U> \<A> \<V> \<B>"
ltl_prop_implies_def: fixes \<phi> :: "'b ltl"   and \<psi> :: "'b ltl" shows "\<phi> \<longrightarrow>\<^sub>P \<psi> \<equiv> \<forall>\<A>. \<A> \<Turnstile>\<^sub>P \<phi> \<longrightarrow> \<A> \<Turnstile>\<^sub>P \<psi>"
cSUP_least: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'c"   and M :: "'c" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "A \<noteq> {}"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq (f x) M" shows "less_eq (Sup (f ` A)) M"
big_preserves_fresh: fixes e :: "trm"   and e' :: "trm"   and x :: "name" assumes "e \<Down> e'"   and "x \<sharp> e" shows "x \<sharp> e'"
pre_4: shows "\<forall>ty\<in>ran (map_of cto). wf_type (map_of cto, map_of tao, translate_osig sa) ty"
Vfrom_mono1: fixes a :: "V"   and b :: "V"   and i :: "V" assumes "a \<subseteq>\<^sub>\<circ> b" shows "Vfrom a i \<subseteq>\<^sub>\<circ> Vfrom b i"
insert_neq_0: fixes a :: "V"   and X :: "V set" shows "(ZFC_in_HOL.set (insert a X) = []\<^sub>\<circ>) = (\<not> small X)"
rep_eq: fixes x :: "'b \<Rightarrow>\<^sub>0 'c"   and xa :: "'b \<Rightarrow>\<^sub>0 'c" shows "lookup (x * xa) = prod_fun (lookup x) (lookup xa)"
not_gr0: fixes n :: "nat" shows "(\<not> order.greater n 0) = (n = 0)"
subadditive_converges_bounded: fixes u :: "nat \<Rightarrow> real" assumes "subadditive u"   and "bdd_below {u n / real n |n. 0 < n}" shows "(\<lambda>n. u n / real n) \<longlonglongrightarrow> Inf {u n / real n |n. 0 < n}"
ie: shows "real_of_int \<lfloor>Inum (real_of_int i # bs) e\<rfloor> = Inum (real_of_int i # bs) e"
NSLIMSEQ_D: fixes X :: "nat \<Rightarrow> 'a"   and L :: "'a"   and N :: "nat star" assumes "X \<longlonglongrightarrow>\<^sub>N\<^sub>S L"   and "N \<in> HNatInfinite" shows "(*f* X) N \<approx> star_of L"
effect_def: fixes c :: "'b Heap"   and h :: "heap"   and h' :: "heap"   and r :: "'b" shows "effect c h h' r = (execute c h = Some (r, h'))"
p_fun_sur_id: fixes x :: "'a" assumes "is_p_fun x"   and "is_sur x" shows "x\<^sup>\<smile> ; x = 1'"
tendsto_mult_right_iff: fixes c :: "'a"   and f :: "'b \<Rightarrow> 'a"   and l :: "'a"   and F :: "'b filter" assumes "c \<noteq> (0::'a)" shows "((\<lambda>x. f x * c) \<longlongrightarrow> l * c) F = (f \<longlongrightarrow> l) F"
wt_blocks: fixes Vs :: "char list list"   and Ts :: "ty list"   and vs :: "val list"   and 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 T :: "ty" assumes "length Vs = length Ts"   and "length vs = length Ts" shows "(P,E,h,sh \<turnstile> blocks (Vs, Ts, vs, e) : T) = (P,E(Vs [\<mapsto>] Ts),h,sh \<turnstile> e : T \<and> (\<exists>Ts'. map typeof\<^bsub>h\<^esub> vs = map Some Ts' \<and> P \<turnstile> Ts' [\<le>] Ts))"
run_raise_iff: fixes s :: "String.literal"   and \<sigma> :: "heap option"   and \<sigma>' :: "heap option"   and r :: "'d" shows "run (raise s) \<sigma> \<sigma>' r = (\<sigma>' = None)"
funof_imp_wp: fixes act :: "('b \<times> 'b) set"   and t :: "'b"   and A :: "'b set" assumes "funof act t \<in> A"   and "single_valued act" shows "t \<in> wp act A"
ordIso_iff_ordLeq: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set" shows "ordIso2 r r' = (ordLeq3 r r' \<and> ordLeq3 r' r)"
t_design_axioms: shows "t_design \<V> \<B> \<k> 2 \<Lambda>"
composite_cell_in_hom: fixes w :: "'a"   and u :: "'a"   and \<theta> :: "'a" assumes "ide w"   and "\<guillemotleft>w : src u \<rightarrow> src f\<guillemotright>"   and "\<guillemotleft>\<theta> : f \<star> w \<Rightarrow> u\<guillemotright>" shows "\<guillemotleft>(r \<star> \<theta>) \<cdot> \<a>[r, f, w] \<cdot> (\<rho> \<cdot> \<phi> \<star> w) : g' \<star> w \<Rightarrow> r \<star> u\<guillemotright>"
lmember_def: fixes x :: "'a"   and xs :: "'a llist" shows "lmember x xs = (x \<in> lset xs)"
pre_ex_rule: fixes P :: "'b \<Rightarrow> assn"   and f :: "'a Heap"   and Q :: "'a \<Rightarrow> assn" shows "(\<not> <\<exists>\<^sub>Ax. P x> f <Q>) = (\<exists>x. \<not> <P x> f <Q>)"
Ln_ii: shows "Ln \<i> = \<i> * complex_of_real pi / 2"
vcomp_eq: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<nu> :: "'b arrow_of_spans_data"   and \<mu> :: "'b arrow_of_spans_data" assumes "span_vertical_category C"   and "partial_magma.arr (span_vertical_category.vcomp C) (span_vertical_category.vcomp C \<nu> \<mu>)" shows "span_vertical_category.vcomp C \<nu> \<mu> = \<lparr>Chn = C (Chn \<nu>) (Chn \<mu>), Dom = Dom \<mu>, Cod = Cod \<nu>\<rparr>"
hadd_succ_right: fixes x :: "hf"   and y :: "hf" shows "x @+ succ y = succ (x @+ y)"
dagger_unfoldl_distr: fixes y :: "'a"   and x :: "'a" shows "y + x \<cdot> x\<^sup>\<dagger> \<cdot> y = x\<^sup>\<dagger> \<cdot> y"
scaleR_left_distrib: fixes a :: "real"   and b :: "real"   and x :: "'b" shows "(a + b) *\<^sub>R x = a *\<^sub>R x + b *\<^sub>R x"
ifmi_les: fixes s :: "bdd"   and v :: "nat"   and ni1 :: "nat"   and ni2 :: "nat"   and ni :: "nat"   and s' :: "bdd" assumes "bdd_sane s"   and "ifmi v ni1 ni2 s = (ni, s')" shows "mi.les s s'"
W'': fixes k :: "nat" assumes "k < length vs" shows "vs ! k \<in> carrier_vec n"
comp_vwb_lens: fixes x :: "'e \<Longrightarrow> 'f"   and y :: "'f \<Longrightarrow> 'g" assumes "vwb_lens x"   and "vwb_lens y" shows "vwb_lens (x ;\<^sub>L y)"
Append_to_free2: fixes i :: "nat"   and e :: "(nat \<times> nat) list"   and n :: "nat" assumes "i \<notin> Reach e" shows "(n \<in> Reach (Append_to_free (i, e))) = (n = i \<or> n \<in> Reach e)"
distinct_disc_14: fixes event :: "('d, 'e, 'f) event" assumes "isRecvMarker event" shows "\<not> isSend event"
pcr_zmultiset_def: fixes T :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "pcr_zmultiset T \<equiv> rel_prod (rel_mset T) (rel_mset T) OO cr_zmultiset"
T4E: fixes T :: "'b set set"   and A :: "'b set"   and B :: "'b set" assumes "T4 T"   and "A closed\<^bsub>T\<^esub>"   and "A \<subseteq> carrier\<^bsub>T\<^esub>"   and "B closed\<^bsub>T\<^esub>"   and "B \<subseteq> carrier\<^bsub>T\<^esub>"   and "A \<inter> B = {}" obtains U :: "'b set" and V :: "'b set" where "U open\<^bsub>T\<^esub>" and "A \<subseteq> U" and "V open\<^bsub>T\<^esub>" and "B \<subseteq> V" and "U \<inter> V = {}"
rl: assumes "Y \<noteq> {}" shows "f \<in> \<surd>ideal F"
tso_store_refs_def: fixes m :: "'a" shows "mut_m.tso_store_refs m = (\<lambda>s. \<Union> (store_refs ` set (sys_mem_store_buffers (mutator m) s)))"
prems: shows "t \<in> pair ` setops\<^sub>s\<^sub>s\<^sub>t (x # S \<cdot>\<^sub>s\<^sub>s\<^sub>t \<theta>)"
assms_2: fixes u :: "'a stream"   and v :: "'b stream" assumes "R u v" shows "\<exists>u\<^sub>1 u\<^sub>2 v\<^sub>1 v\<^sub>2. u = u\<^sub>1 @- u\<^sub>2 \<and> v = v\<^sub>1 @- v\<^sub>2 \<and> u\<^sub>1 \<noteq> [] \<and> v\<^sub>1 \<noteq> [] \<and> list_all2 P u\<^sub>1 v\<^sub>1 \<and> R u\<^sub>2 v\<^sub>2"
predicate2D_conj: fixes P :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R :: "bool"   and x :: "'d"   and y :: "'e" assumes "P \<le> Q \<and> R" shows "R \<and> (P x y \<longrightarrow> Q x y)"
shextI: fixes sh :: "sheap"   and sh' :: "sheap" assumes "\<forall>C sfs i. sh C = \<lfloor>(sfs, i)\<rfloor> \<longrightarrow> (\<exists>sfs' i'. sh' C = \<lfloor>(sfs', i')\<rfloor> \<and> i \<le>\<^sub>i i')" shows "sh \<unlhd>\<^sub>s sh'"
C11_var: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and star :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.conway_dioid plus times one zero less_eq less star" shows "star (plus x y) = times (star x) (star (times y (star x)))"
assms: shows "gen_set {\<zero>\<^bsub>V\<^esub>}"
eval_runiq_rel: fixes x :: "'a"   and R :: "('a \<times> 'b) set" assumes "x \<in> Domain R"   and "runiq R" shows "(x, R ,, x) \<in> R"
atLeastLessThan_eq_atLeastAtMost_diff: fixes a :: "'a"   and b :: "'a" shows "{a..<b} = {a..b} - {b}"
abs_eq: fixes x :: "'a \<times> 'a scheduler" shows "state (abs_cfg x) = fst x"
wordinterval_compress: fixes r :: "'a wordinterval" shows "wordinterval_to_set (wordinterval_compress r) = wordinterval_to_set r"
Sigma_fm_AND: fixes A :: "fm"   and B :: "fm" assumes "Sigma_fm A"   and "Sigma_fm B" shows "Sigma_fm (A AND B)"
last_rev: fixes xs :: "'g list" assumes "xs \<noteq> []" shows "last (rev xs) = hd xs"
eq1: fixes i :: "nat" shows "LBINT x=ereal (l i)..ereal (u i). f (g x) * g' x = interval_lebesgue_integral lborel (ereal (g (l i))) (ereal (g (u i))) f"
grid_change_dim_child: fixes p :: "(nat \<times> int) list"   and b :: "(nat \<times> int) list"   and ds :: "nat set"   and d :: "nat"   and dir :: "dir" assumes "p \<in> grid b ds"   and "d \<notin> ds" shows "child p dir d \<in> grid (child b dir d) ds"
cfg_on_not_empty: fixes K :: "'a \<Rightarrow> 'a pmf set"   and s :: "'a" assumes "Markov_Decision_Process K" shows "Markov_Decision_Process.cfg_on K s \<noteq> {}"
fls_divide_subdegree_ge: fixes f :: "'a fls"   and g :: "'a fls" assumes "f / g \<noteq> 0" shows "fls_subdegree f - fls_subdegree g \<le> fls_subdegree (f / g)"
degree_smult_not_zero_divisor_right: fixes p :: "'b poly"   and c :: "'b" assumes "\<not> zero_divisor (lead_coeff p)" shows "degree (smult c p) = (if c = (0::'b) then 0 else degree p)"
refl: fixes x :: "(('a, 'b) T\<^sub>m, 'b T\<^sub>h, 'c, 'd) F\<^sub>m" assumes "x \<in> {x. set1_F\<^sub>m x \<subseteq> {x. set1_T\<^sub>m x \<subseteq> A} \<and> set3_F\<^sub>m x \<subseteq> B}" shows "blinding_of_F (root_hash_T rha) (blinding_of_T rha boa) rhb bob x x"
bdd_sane_def: fixes bdd :: "bdd" shows "bdd_sane bdd \<equiv> pointermap_sane (dpm bdd) \<and> mi_pre.map_invar_impl (dcl bdd) bdd"
simps_2: fixes xs :: "'a list" shows "shuffles xs [] = {xs}"
subspace_scale: fixes Sa :: "'a set"   and x :: "'a"   and c :: "real" assumes "Sa \<subseteq> S"   and "subspace Sa"   and "x \<in> Sa" shows "c *\<^sub>R x \<in> Sa"
new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t: fixes h :: "(_) heap"   and new_shadow_root_ptr :: "(_) shadow_root_ptr"   and h' :: "(_) heap"   and ptr :: "(_) document_ptr" assumes "new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t h = (new_shadow_root_ptr, h')"   and "ptr \<noteq> cast new_shadow_root_ptr" shows "get ptr h = get ptr h'"
EmptyMap_Root: fixes SA :: "('d, 'e, 'f) seqauto" shows "Root {SA} (EmptyMap (States SA)) = SA"
nonzero_octo_of_real_inverse: fixes x :: "real" assumes "x \<noteq> 0" shows "octo_of_real (inverse x) = inverse (octo_of_real x)"
le_iff_add: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "less_eq a b = (\<exists>c. b = plus a c)"
th3_7_1: fixes b :: "(real, 'b) vec"   and c :: "(real, 'b) vec"   and a :: "(real, 'b) vec"   and d :: "(real, 'b) vec" assumes "b \<noteq> c"   and "B\<^sub>\<real> a b c"   and "B\<^sub>\<real> b c d" shows "B\<^sub>\<real> a c d"
cong_altdef_nat': fixes a :: "nat"   and b :: "nat"   and m :: "nat" shows "[a = b] (mod m) = (m dvd nat \<bar>int a - int b\<bar>)"
assms_1: shows "finite (reach\<^sub>t \<Sigma> \<delta> q\<^sub>0)"
nth_aforms_errI: fixes vs :: "real list"   and e :: "nat \<Rightarrow> real"   and VS :: "((real \<times> real pdevs) \<times> real) list"   and i :: "nat" assumes "vs \<in> aforms_err e VS"   and "i < length vs" shows "vs ! i \<in> aform_err e (VS ! i)"
simps_1: fixes j :: "nat"   and k :: "nat" shows "freedb (DBVar j) k = (j = k)"
getChoices_sub_edges: fixes a :: "edge"   and b :: "pState"   and g :: "gState\<^sub>I"   and p :: "pState"   and es :: "edge list" assumes "(a, b) \<in> set (getChoices g p es)" shows "a \<in> set es"
gauss_int_2_eq: shows "2 = - \<i>\<^sub>\<int> * (1 + \<i>\<^sub>\<int>)\<^sup>2"
that: fixes x :: "V" assumes "x \<in>\<^sub>\<circ> a"   and "x \<inter>\<^sub>\<circ> a = []\<^sub>\<circ>" shows "thesis"
\<S>\<^sub>G_s: shows "\<S>\<^sub>G s"
mulex1_singleton: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" shows "mulex1 P {#x#} {#y#} = P x y"
sum_div_card: fixes f :: "nat \<Rightarrow> real"   and n :: "nat"   and q :: "real" shows "sum f {..<n} + q * real n = (\<Sum>l<n. f l + q)"
ran_m_mapsto_upd: fixes C :: "'a"   and N :: "('a, 'b) fmap"   and C' :: "'b" assumes "C \<in># dom_m N" shows "ran_m (fmupd C C' N) = add_mset C' (remove1_mset (the (fmlookup N C)) (ran_m N))"
sel_3: fixes x31 :: "nat"   and x32 :: "'d"   and x33 :: "'d"   and x34 :: "'e"   and x35 :: "'e"   and x36 :: "'f" shows "occurs_on (Recv x31 x32 x33 x34 x35 x36) = x32"
range: fixes A :: "'a set" shows "local.MaxR.MaxR_opt A \<in> Some ` (A \<inter> Field r) \<union> {None}"
subseq_append: fixes xs :: "'a list"   and zs :: "'a list"   and ys :: "'a list" shows "(xs @ zs \<preceq> ys @ zs) = (xs \<preceq> ys)"
set_eq_sym: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> carrier L"   and "B \<subseteq> carrier L"   and "A {.=} B" shows "B {.=} A"
sets_image_in_sets: fixes N :: "'c measure"   and X :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and M :: "'d measure" assumes "space N = X"   and "f \<in> N \<rightarrow>\<^sub>M M" shows "sets (vimage_algebra X f M) \<subseteq> sets N"
satisfies_bounded_norm: fixes \<AA> :: "'i"   and \<phi> :: "('a, 'k) aformula" shows "(\<AA> \<Turnstile>\<^sub>b norm \<phi>) = (\<AA> \<Turnstile>\<^sub>b \<phi>)"
mem_in_image: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'f set"   and a :: "'e" assumes "f \<in> A \<rightarrow> B"   and "a \<in> A" shows "f a \<in> f ` A"
Diag_Diagonalize_3: fixes t :: "'a local.term" assumes "Arr t" shows "Cod \<^bold>\<lfloor>t\<^bold>\<rfloor> = \<^bold>\<lfloor>Cod t\<^bold>\<rfloor>"
these_UNION: fixes B :: "'c \<Rightarrow> 'b option set"   and A :: "'c set" shows "Option.these (\<Union> (B ` A)) = \<Union> ((Option.these \<circ> B) ` A)"
deriv_f: fixes s :: "(nat \<times> form) list" shows "deriv s = (\<Union>x. set (map (Pair x) (f [s] x)))"
mnext: fixes I :: "\<I>"   and xs :: "event_data option list set list"   and ts :: "nat list"   and ys :: "event_data option list set list"   and xs' :: "event_data option list set list"   and ts' :: "nat list"   and P :: "nat \<Rightarrow> event_data option list set \<Rightarrow> bool"   and i :: "nat"   and j' :: "nat"   and \<sigma> :: "'a trace"   and j :: "nat" assumes "mprev_next I xs ts = (ys, xs', ts')"   and "list_all2 P [Suc i..<j'] xs"   and "list_all2 (\<lambda>i t. t = \<tau> \<sigma> i) [i..<j] ts"   and "Suc i \<le> j'"   and "i < j" shows "list_all2 (\<lambda>i X. if mem (\<tau> \<sigma> (Suc i) - \<tau> \<sigma> i) I then P (Suc i) X else X = empty_table) [i..<min (j' - 1) (j - 1)] ys \<and> list_all2 P [Suc (min (j' - 1) (j - 1))..<j'] xs' \<and> list_all2 (\<lambda>i t. t = \<tau> \<sigma> i) [min (j' - 1) (j - 1)..<j] ts'"
corSim: shows "weak_reduction_correspondence_simulation Rel (STCal Source Target)"
StableD: fixes F :: "'d program"   and A :: "'d set" assumes "F \<in> Stable A" shows "F \<in> A Co A"
store_sops_le: fixes i :: "nat"   and s :: "stmt"   and t :: "nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "(stmt \<times> nat) memref list"   and \<D> :: "'a"   and \<O> :: "'b" assumes "i < length ts"   and "ts ! i = ((s, t), is, \<theta>, sb, \<D>, \<O>)" shows "\<forall>i\<in>\<Union> (fst ` store_sops is). i < t"
absorb2: fixes a :: "'a Word.word"   and b :: "'a Word.word" assumes "a \<le>s b" shows "signed.max a b = b"
indep_sets_mono_index: fixes J :: "'b set"   and I :: "'b set"   and F :: "'b \<Rightarrow> 'a set set" assumes "J \<subseteq> I"   and "indep_sets F I" shows "indep_sets F J"
merge_in_prod_emb: fixes y :: "'d \<Rightarrow> 'e"   and I :: "'d set"   and M :: "'d \<Rightarrow> 'e measure"   and x :: "'d \<Rightarrow> 'e"   and X :: "('d \<Rightarrow> 'e) set"   and J :: "'d set" assumes "y \<in> space (Pi\<^sub>M I M)"   and "x \<in> X"   and "X \<in> sets (Pi\<^sub>M J M)"   and "J \<subseteq> I" shows "merge J I (x, y) \<in> prod_emb I M J X"
that: shows "y - b \<in> aform_err e ((a, xs), d)"
set_discriminating_def: fixes n :: "nat"   and f :: "bool stream \<Rightarrow> 'b"   and N :: "'b measure" shows "set_discriminating n f N \<equiv> \<forall>w. f w \<noteq> f (pseudo_proj_True n w) \<longrightarrow> (\<exists>A\<in>sets N. (f w \<in> A) = (f (pseudo_proj_True n w) \<notin> A))"