hcju/isabelleps huggingface dataset:
cas_ends: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and p :: "'d list"   and v :: "'c"   and u' :: "'c"   and v' :: "'c" assumes "wf_digraph G"   and "pre_digraph.cas G u p v"   and "pre_digraph.cas G u' p v'" shows "p \<noteq> [] \<and> u = u' \<and> v = v' \<or> p = [] \<and> u = v \<and> u' = v'"
space_tail_algebra: fixes M :: "'b measure"   and n :: "nat" assumes "shift_operator_ergodic M" shows "space (shift_operator_ergodic.tail_algebra M n) = UNIV \<rightarrow>\<^sub>E space M"
monad_gpv: shows "monad Generative_Probabilistic_Value.Done (\<bind>)"
list_of_llist_of: fixes xs :: "'a list" shows "list_of (llist_of xs) = xs"
count_replicate_mset: fixes n :: "nat"   and x :: "'c"   and y :: "'c" shows "count (replicate_mset n x) y = (if y = x then n else 0)"
dvd_Gcd_iff: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and Gcd :: "'a set \<Rightarrow> 'a"   and Lcm :: "'a set \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and x :: "'a"   and A :: "'a set" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize" shows "dvd.dvd times x (Gcd A) = (\<forall>y\<in>A. dvd.dvd times x y)"
comp_ide_source: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and f :: "'b" assumes "regular_weak_composition V H"   and "a \<in> weak_composition.sources V H f"   and "partial_magma.ide V f" shows "category.isomorphic V (H f a) f"
measurable_prob_algebraD: fixes f :: "'c \<Rightarrow> 'd measure"   and N :: "'c measure"   and M :: "'d measure" assumes "f \<in> N \<rightarrow>\<^sub>M prob_algebra M" shows "f \<in> N \<rightarrow>\<^sub>M subprob_algebra M"
prems_4: fixes xs :: "'a list"   and ys :: "'b list"   and x :: "'a"   and y :: "'b" assumes "P xs ys" shows "P (xs \<cdot> [x]) (ys \<cdot> [y])"
arity_primerec: fixes f :: "recf"   and n :: "nat" assumes "primerec f n" shows "arity f = n"
space_completion: fixes M :: "'c measure" shows "space (completion M) = space M"
c_f_pair_to_pr: fixes f :: "nat \<Rightarrow> nat"   and g :: "nat \<Rightarrow> nat" assumes "f \<in> PrimRec1"   and "g \<in> PrimRec1" shows "c_f_pair f g \<in> PrimRec1"
substA_congA: fixes at :: "('c, 'd) atm"   and \<pi>1 :: "Preliminaries.var \<Rightarrow> 'c trm"   and \<pi>2 :: "Preliminaries.var \<Rightarrow> 'c trm" assumes "\<And>x. x \<in> varsA at \<Longrightarrow> \<pi>1 x = \<pi>2 x" shows "substA \<pi>1 at = substA \<pi>2 at"
subdprodr_impl_def: fixes a :: "nat list"   and b :: "nat list"   and ys :: "nat list" shows "subdprodr_impl a b ys = (\<forall>l\<le>length b. take l b \<bullet> take l ys \<le> a \<bullet> map (max_x_impl a b (take l ys)) [0..<length a])"
Ks'_2: shows "set Ks' \<subseteq> set (take (length Us) Ks) \<union> {\<zero>}"
size_10: fixes x41 :: "'b formula"   and x42 :: "'b formula" shows "size (x41 \<^bold>\<and> x42) = size x41 + size x42 + Suc 0"
minus_closed: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "y \<in> carrier Zp" shows "x \<ominus> y \<in> carrier Zp"
BRANCH_not_LINEAR: fixes c :: "'f"   and rs :: "'i \<Rightarrow> ('f + 'g + 'h, 'i) Old_Datatype.node set"   and b :: "'g"   and r :: "('f + 'g + 'h, 'i) Old_Datatype.node set" shows "BRANCH c rs \<noteq> LINEAR b r"
B: fixes x :: "complex" assumes "x \<in> path_image \<gamma>'" shows "cmod x \<le> B"
low_indistinguishable_def: fixes mds :: "Mode \<Rightarrow> 'Var set"   and c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt" shows "c\<^sub>1 \<sim>\<^bsub>mds\<^esub> c\<^sub>2 = (\<forall>mem\<^sub>1 mem\<^sub>2. mem\<^sub>1 =\<^bsub>mds\<^esub>\<^sup>l mem\<^sub>2 \<longrightarrow> \<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<approx> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>)"
neg_diag_empty_spec: fixes i :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'b DBMEntry" assumes "i \<le> n"   and "M i i < \<one>" shows "[M]\<^bsub>v,n\<^esub> = {}"
CallRec: fixes P :: "'e set"   and p :: "'f"   and Q :: "'e set"   and A :: "'e set"   and Specs :: "('e set \<times> 'f \<times> 'e set \<times> 'e set) set"   and r :: "(('e \<times> 'f) \<times> 'e \<times> 'f) set"   and Specs_wf :: "'f \<Rightarrow> 'e \<Rightarrow> ('e set \<times> 'f \<times> 'e set \<times> 'e set) set"   and \<Gamma> :: "'f \<Rightarrow> ('e, 'f, 'g) com option"   and \<Theta> :: "('e set \<times> 'f \<times> 'e set \<times> 'e set) set"   and F :: "'g set" assumes "(P, p, Q, A) \<in> Specs"   and "wf r"   and "Specs_wf = (\<lambda>p \<sigma>. (\<lambda>(P, q, Q, A). (P \<inter> {s. ((s, q), \<sigma>, p) \<in> r}, q, Q, A)) ` Specs)"   and "\<forall>(P, p, Q, A)\<in>Specs. p \<in> dom \<Gamma> \<and> (\<forall>\<sigma>. \<Gamma>,\<Theta> \<union> Specs_wf p \<sigma>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> ({\<sigma>} \<inter> P) the (\<Gamma> p) Q,A)" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P Call p Q,A"
finfun_Diag_def: fixes f :: "('a, 'b) finfun"   and g :: "('a, 'c) finfun" shows "finfun_Diag f g = finfun_rec (\<lambda>b. finfun_comp (Pair b) g) (\<lambda>a b c. finfun_update c a (b, finfun_apply g a)) f"
PE_arg_rel_def: shows "PE_arg_rel \<equiv> \<lambda>i. ValF\<cdot>(case i of 0 \<Rightarrow> \<Lambda> uu_. ValFF | Suc n \<Rightarrow> \<Lambda> (ValN\<cdot>x). if x = Suc n then ValTT else \<bottom>)"
case_eq_if: fixes f1 :: "('b, 'c) hd \<Rightarrow> 'a"   and f2 :: "('b, 'c) tm \<Rightarrow> ('b, 'c) tm \<Rightarrow> 'a"   and tm :: "('b, 'c) tm" shows "(case tm of Hd x \<Rightarrow> f1 x | App x xa \<Rightarrow> f2 x xa) = (if is_Hd tm then f1 (head tm) else f2 (fun tm) (arg tm))"
SUP_UNION: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'f \<Rightarrow> 'e set"   and A :: "'f set" shows "\<Squnion> (f ` \<Union> (g ` A)) = (\<Squnion>y\<in>A. \<Squnion> (f ` g y))"
pp_dist_inf: fixes x :: "'c"   and y :: "'c" shows "- - (x \<squnion> y) = - - x \<squnion> - - y"
cont_on_top: shows "continuous_on (path_image (\<lambda>x. twoC (1, x))) F"
convex_imp_AR: fixes S :: "'a set" assumes "convex S"   and "S \<noteq> {}" shows "AR S"
index_nth_id: fixes xs :: "'c list"   and n :: "nat" assumes "distinct xs"   and "n < length xs" shows "index xs (xs ! n) = n"
ft_invar_def: fixes t :: "('c, 'd) FingerTreeStruc" shows "ft_invar t \<equiv> is_leveln_ftree 0 t \<and> is_measured_ftree t"
ring_hom_ring_axioms_def: fixes R :: "('e, 'f) ring_scheme"   and S :: "('g, 'h) ring_scheme"   and h :: "'e \<Rightarrow> 'g" shows "ring_hom_ring_axioms R S h \<equiv> h \<in> ring_hom R S"
intros_4: fixes \<sigma> :: "'var \<Rightarrow> 'val"   and k :: "nat"   and \<sigma>' :: "'var \<Rightarrow> 'val"   and k' :: "nat" assumes "((\<sigma>, k), \<sigma>', k') \<in> cp" shows "((\<sigma>', k'), \<sigma>, k) \<in> cp"
remap_ta: fixes f :: "'Q \<Rightarrow> 'b" shows "tree_automaton (ta_remap f TA)"
A: shows "MASTER_BOUND p' 0 0 \<in> \<Theta>(\<lambda>x. real x powr p')"
diff_conv_add_uminus: fixes a :: "'c"   and b :: "'c" shows "a - b = a + - b"
join_right: fixes y :: "'a"   and x :: "'a" shows "y \<sqsubseteq> x \<squnion> y"
IH: fixes y :: "nat" assumes "y < n"   and "0 < y" shows "F y < 4 ^ y"
unit_eqI: fixes a :: "'b"   and \<mu> :: "'b" assumes "B.obj a"   and "\<guillemotleft>\<mu> : I.map\<^sub>0 a \<rightarrow>\<^sub>B local.map a\<guillemotright>"   and "B.iso \<mu>"   and "\<mu> \<cdot>\<^sub>B \<i>\<^sub>B[I.map\<^sub>0 a] = (local.map \<i>\<^sub>B[a] \<cdot>\<^sub>B cmp (a, a)) \<cdot>\<^sub>B (\<mu> \<star>\<^sub>B \<mu>)" shows "\<mu> = unit a"
hh: shows "dist_perm ((swap 0 ^^ ALG_P ss x y s1) s2) ((swap 0 ^^ ALG_P ss x y s1) s2)"
match_def: fixes step :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<times> 'a"   and \<phi> :: "('a, 'b, 'c) trans \<Rightarrow> bool"   and f :: "('a, 'b, 'c) trans \<Rightarrow> 'd"   and \<gamma> :: "('a, 'b, 'c) trans \<Rightarrow> bool"   and g :: "('a, 'b, 'c) trans \<Rightarrow> 'e"   and \<Delta> :: "'a \<Rightarrow> 'f \<Rightarrow> 'a \<Rightarrow> 'd list \<Rightarrow> bool"   and s :: "'a"   and s1 :: "'a"   and vl1 :: "'d list"   and a :: "'b"   and ou :: "'c"   and s' :: "'a"   and vl' :: "'f" shows "BD_Security.match step \<phi> f \<gamma> g \<Delta> s s1 vl1 a ou s' vl' \<equiv> \<exists>a1 ou1 s1' vl1'. let trn = Trans s a ou s'; trn1 = Trans s1 a1 ou1 s1' in IO_Automaton.validTrans step trn1 \<and> BD_Security.consume \<phi> f trn1 vl1 vl1' \<and> \<gamma> trn = \<gamma> trn1 \<and> (\<gamma> trn \<longrightarrow> g trn = g trn1) \<and> \<Delta> s' vl' s1' vl1'"
Cons_2: shows "mset (y # ys) \<subseteq># mset xs"
execn_to_exec: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate"   and n :: "nat"   and t :: "('b, 'c) xstate" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t"
raw_filtermap_apply: fixes f :: "'d \<Rightarrow> 'a"   and F :: "('d \<Rightarrow> 'b) \<Rightarrow> 'c" shows "raw_filtermap f F = (\<lambda>P. F (\<lambda>x. P (f x)))"
successor_set_finite: fixes n :: "'node" assumes "valid_node n" shows "finite {n'. \<exists>a'. valid_edge a' \<and> sourcenode a' = n \<and> targetnode a' = n'}"
PRelQ: shows "P \<leadsto>\<^sup>^<Rel> Q"
prems_1: shows "set (m # ms) \<subseteq> M"
prime_factorization_subset_iff_dvd: fixes x :: "'a"   and y :: "'a" assumes "x \<noteq> (0::'a)"   and "y \<noteq> (0::'a)" shows "(prime_factorization x \<subseteq># prime_factorization y) = (x dvd y)"
k_def: fixes G1 :: "'b poly"   and G2 :: "'b poly" shows "??.Subresultant.k G1 G2 = (LEAST i. ??.Subresultant.F G1 G2 (Suc i) = 0)"
simps_5: fixes v :: "'b" shows "(\<infinity> \<le> Fin v) = False"
postdominate_antisym: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> 'c edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and Exit :: "'b"   and n' :: "'b"   and n :: "'b" assumes "Postdomination sourcenode targetnode kind valid_edge Entry Exit"   and "Postdomination.postdominate sourcenode targetnode valid_edge Exit n' n"   and "Postdomination.postdominate sourcenode targetnode valid_edge Exit n n'" shows "n = n'"
char_one_nz: shows "dc2c \<chi> \<one>\<^bsub>G\<^esub> \<noteq> 0"
rep_eq: fixes x :: "complex"   and xa :: "'a \<Rightarrow>\<^sub>L 'b" shows "(*\<^sub>v) (x *\<^sub>C xa) = (\<lambda>xb. x *\<^sub>C (xa *\<^sub>v xb))"
case_prodD: fixes f :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and a :: "'k"   and b :: "'l" assumes "case (a, b) of (c, d) \<Rightarrow> f c d" shows "f a b"
bind_coin_spmf_eq_const: fixes b :: "bool" shows "coin_spmf \<bind> (\<lambda>x. return_spmf (b = x)) = coin_spmf"
nonneg_Reals_divide_I: fixes a :: "'a"   and b :: "'a" assumes "a \<in> \<real>\<^sub>\<ge>\<^sub>0"   and "b \<in> \<real>\<^sub>\<ge>\<^sub>0" shows "a / b \<in> \<real>\<^sub>\<ge>\<^sub>0"
simps_3: fixes a :: "'b" shows "lang (Atom a) = {[a]}"
a_distrib2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "ad x \<oplus> ad y \<cdot> ad z = (ad x \<oplus> ad y) \<cdot> (ad x \<oplus> ad z)"
bounded_imp_bdd_below: fixes S :: "real set" assumes "bounded S" shows "bdd_below S"
int: shows "\<integral>\<^sup>+ x. ennreal (indicat_real {0<..} x * (1 + x\<^sup>2) powr a) \<partial>lborel = ennreal C"
one_closed: shows "\<one>\<^bsub>ideals_set R\<^esub> \<in> carrier (ideals_set R)"
pre_iso_test3: shows "\<forall>g\<in>set Tri. pre_iso_test g"
commutativity1_lemma: fixes F :: "'b program"   and T :: "'b set"   and B :: "'b set"   and L :: "'b set set" assumes "commutes F T B L"   and "lattice L"   and "B \<in> L"   and "T \<in> L" shows "closed F T B L"
abs_eq: shows "0 = toQ \<lparr>mag = 0::'a, dim = QD('b), unit_sys = unit\<rparr>"
save_retore_sub1_def: fixes result :: "word32"   and new_cwp :: "word5"   and rd :: "word5" shows "save_retore_sub1 result new_cwp rd \<equiv> do psr_val \<leftarrow> gets (cpu_reg_val PSR); new_psr_val \<leftarrow> gets (\<lambda>s. update_CWP new_cwp psr_val); write_cpu new_psr_val PSR; write_reg result (ucast new_cwp) rd; return () od"
select_convs_4: fixes getSATFlag :: "ExtendedBool"   and getF :: "Literal list list"   and getM :: "(Literal \<times> bool) list"   and getConflictFlag :: "bool"   and getConflictClause :: "nat"   and getQ :: "Literal list"   and getReason :: "Literal \<Rightarrow> nat option"   and getWatch1 :: "nat \<Rightarrow> Literal option"   and getWatch2 :: "nat \<Rightarrow> Literal option"   and getWatchList :: "Literal \<Rightarrow> nat list"   and getC :: "Literal list"   and getCl :: "Literal"   and getCll :: "Literal"   and getCn :: "nat"   and more :: "'a" shows "getConflictFlag \<lparr>getSATFlag = getSATFlag, getF = getF, getM = getM, getConflictFlag = getConflictFlag, getConflictClause = getConflictClause, getQ = getQ, getReason = getReason, getWatch1 = getWatch1, getWatch2 = getWatch2, getWatchList = getWatchList, getC = getC, getCl = getCl, getCll = getCll, getCn = getCn, \<dots> = more\<rparr> = getConflictFlag"
nth_Abs_fls_ex_lower_bound: fixes f :: "int \<Rightarrow> 'a"   and n :: "int" assumes "\<exists>N. \<forall>n<N. f n = (0::'a)" shows "fls_nth (Abs_fls f) n = f n"
ord_option_compp: fixes A :: "'c \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'd \<Rightarrow> bool" shows "ord_option (A OO B) = ord_option A OO ord_option B"
swap_swapped: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and zs :: "'varSort"   and x :: "'var"   and y :: "'var" shows "(X, X #[x \<and> y]_zs) \<in> swapped"
Nil: fixes G :: "prog"   and s0 :: "st"   and n :: "nat" shows "G\<turnstile>Norm s0 \<midarrow>[]\<doteq>\<succ>[]\<midarrow>n\<rightarrow> Norm s0"
\<tau>_def: fixes n :: "nat"   and z :: "int" shows "\<tau> n z = mset (map (\<lambda>i. (f ^^ nat i) z) [0..int n - 1])"
rbt_sorted_fold_insert: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and t :: "('b, 'c) rbt"   and t' :: "('b, 'c) rbt" assumes "class.linorder less_eq less"   and "ord.rbt_sorted less t" shows "ord.rbt_sorted less (RBT_Impl.fold (ord.rbt_insert less) t' t)"
bres_galois_imp: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<cdot> y \<le> z \<longrightarrow> y \<le> x \<rightarrow> z"
inf_right_idem: fixes x :: "'a multiset"   and y :: "'a multiset" shows "x \<inter># y \<inter># y = x \<inter># y"
product_run_project_limit: fixes \<iota>\<^sub>m :: "'b \<Rightarrow> 'a option"   and x :: "'b"   and q\<^sub>0 :: "'a"   and \<delta>\<^sub>m :: "'b \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> 'a"   and w :: "nat \<Rightarrow> 'c" assumes "\<iota>\<^sub>m x = Some q\<^sub>0"   and "finite (range (run\<^sub>t (\<Delta>\<^sub>\<times> \<delta>\<^sub>m) \<iota>\<^sub>m w))" shows "\<downharpoonleft>\<^sub>x ` limit (run\<^sub>t (\<Delta>\<^sub>\<times> \<delta>\<^sub>m) \<iota>\<^sub>m w) = limit (run\<^sub>t (\<delta>\<^sub>m x) q\<^sub>0 w)"
machin: shows "pi / 4 = 4 * arctan (1 / 5) - arctan (1 / 239)"
assms: shows "subseq as1 (rem_effectless_act as2)"
trans: fixes b :: "'b"   and a :: "'b"   and c :: "'b" assumes "b \<le> a"   and "c \<le> b" shows "c \<le> a"
compose_def: fixes A :: "'d set"   and g :: "'f \<Rightarrow> 'e"   and f :: "'d \<Rightarrow> 'f" shows "compose A g f = (\<lambda>x\<in>A. g (f x))"
assms: fixes F :: "unit + 'a \<times> 'a list_R1" assumes "\<And>l'. l' \<in> setr F \<bind> snds \<Longrightarrow> P l'" shows "P (list_R1 F)"
Step1_2_1: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> [HNext rmhist p]_(c p, r p, m p, rmhist!p) \<and> ImpNext p \<and> \<not> unchanged (e p, c p, r p, m p, rmhist!p) \<and> $S1 rmhist p \<longrightarrow> S2 rmhist p$ \<and> ENext p \<and> unchanged (c p, r p, m p)"
bigo_const_iff: fixes c1 :: "'b"   and F :: "'a filter"   and c2 :: "'b" shows "((\<lambda>_. c1) \<in> O[F](\<lambda>_. c2)) = (F = bot \<or> c1 = (0::'b) \<or> c2 \<noteq> (0::'b))"
root_order_imp_divmuleq: fixes k :: "nat"   and t :: "'b btree" assumes "root_order k t" shows "nodes t * k div k = nodes t"
comm_neutral: fixes a :: "'a::refinement_lattice" shows "(a::'a::refinement_lattice) \<iinter> chaos = a"
exists_bezout_extended: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and S :: "'b set" assumes "class.bezout_ring times one plus zero minus uminus"   and "finite S"   and "S \<noteq> {}" shows "\<exists>f d. comm_monoid_add.sum plus zero (\<lambda>a. times (f a) a) S = d \<and> (\<forall>a\<in>S. dvd.dvd times d a) \<and> (\<forall>d'. (\<forall>a\<in>S. dvd.dvd times d' a) \<longrightarrow> dvd.dvd times d' d)"
GS_rel_def: shows "GS_rel \<equiv> br GS.\<alpha> GS_invar"
bij_prop0: fixes g :: "'a" assumes "g \<in> carrier G" shows "\<phi> g \<in> Bij E"
hoaret_sound: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A" shows "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
wt_s_eq_\<delta>: shows "eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (wt s)) = zhmset_of \<delta>\<^sub>h"
exp_conj_bounded_by: fixes P :: "'c \<Rightarrow> real"   and Q :: "'c \<Rightarrow> real" assumes "bounded_by 1 P"   and "bounded_by 1 Q" shows "bounded_by 1 (P && Q)"
simps: fixes c :: "'a" shows "the_In3 (In3 c) = c"
offs_equal: fixes i :: "nat"   and j :: "nat"   and ns :: "nat list"   and m :: "nat" assumes "i < j"   and "j < length ns"   and "\<And>k. k \<in> {i..<j} \<Longrightarrow> ns ! k = 0" shows "offs ns m ! i = offs ns m ! j"
uses'_phis'_def: fixes g :: "'g" shows "uses'_phis' g = (let (u, p) = foldr (aux_2 g) (\<alpha>n g) (Mapping.empty, Mapping.empty) in (u, map_keys (\<lambda>(n, v). (n, v, n, PhiDef)) p))"
w\<^sub>\<tau>: shows "ide (\<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<and> is_left_adjoint (\<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)"
f_Exec_State_Idle_append_replicate_NoMsg_state: fixes localState :: "'d \<Rightarrow> 'e"   and output_fun :: "'d \<Rightarrow> 'f message_af"   and trans_fun :: "'g message_af \<Rightarrow> 'd \<Rightarrow> 'd"   and xs :: "'g message_af list"   and c :: "'d"   and n :: "nat" assumes "State_Idle localState output_fun trans_fun (localState (f_Exec_Comp trans_fun xs c))" shows "localState (f_Exec_Comp trans_fun (xs @ NoMsg\<^bsup>n\<^esup>) c) = localState (f_Exec_Comp trans_fun xs c)"
restrict_on_source: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set"   and T :: "'c set" assumes "Set_Theory.map f S T" shows "restrict f S = f"
simps_1: fixes P :: "'c \<Rightarrow> bool" shows "filter P [] = []"
that: fixes a :: "'c"   and \<chi> :: "'j \<Rightarrow> 'c" assumes "local.limit_cone a \<chi>" shows "thesis"
log_less_iff: fixes b :: "real"   and x :: "real"   and y :: "real" assumes "1 < b"   and "0 < x" shows "(log b x < y) = (x < b powr y)"
l11_18_2: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and C :: "'a"   and B :: "'a"   and D :: "'a"   and A :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Bet C B D"   and "Tarski_neutral_dimensionless.CongA Bet Cong A B C A B D" shows "Tarski_neutral_dimensionless.Per Bet Cong A B C"
diff_fun_space_def: shows "l.diff_fun_space = {f. diff_fun l charts f \<and> extensional0 carrier f}"
HPhase0Read_HInv4c_p: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk" assumes "HPhase0Read s s' p d"   and "HInv4c s p" shows "HInv4c s' p"
mk_until_semantics: fixes w :: "nat \<Rightarrow> 'a set"   and x :: "'a ltln"   and y :: "'a ltln" shows "w \<Turnstile>\<^sub>n mk_until x y = w \<Turnstile>\<^sub>n x U\<^sub>n y"
divide_less_eq_1_pos: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero a" shows "less (divide b a) one = less b a"
Subst_imp_hfthm: fixes e :: "(name, hf) finfun"   and A :: "fm"   and y :: "hf" assumes "Subst \<lbrakk>\<guillemotleft>A\<guillemotright>\<rbrakk>e y"   and "{} \<turnstile> A" shows "\<exists>B. y = \<lbrakk>\<guillemotleft>B\<guillemotright>\<rbrakk>e \<and> {} \<turnstile> B"
split: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "('d, 'e) Term.term \<Rightarrow> 'c"   and f2 :: "('d, 'e) Term.term \<Rightarrow> 'c"   and f3 :: "poscheckvariant \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> 'c"   and f4 :: "('d, 'e) Term.term \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> 'c"   and f5 :: "('d, 'e) Term.term \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> 'c"   and f6 :: "poscheckvariant \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> 'c"   and f7 :: "'e list \<Rightarrow> (('d, 'e) Term.term \<times> ('d, 'e) Term.term) list \<Rightarrow> (('d, 'e) Term.term \<times> ('d, 'e) Term.term) list \<Rightarrow> 'c"   and stateful_strand_step :: "('d, 'e) stateful_strand_step" shows "P (case stateful_strand_step of send\<langle>x\<rangle> \<Rightarrow> f1 x | receive\<langle>x\<rangle> \<Rightarrow> f2 x | \<langle>x: xa \<doteq> xb\<rangle> \<Rightarrow> f3 x xa xb | insert\<langle>x,xa\<rangle> \<Rightarrow> f4 x xa | delete\<langle>x,xa\<rangle> \<Rightarrow> f5 x xa | \<langle>x: xa \<in> xb\<rangle> \<Rightarrow> f6 x xa xb | \<forall>x\<langle>\<or>\<noteq>: xa \<or>\<notin>: xb\<rangle> \<Rightarrow> f7 x xa xb) = ((\<forall>x1. stateful_strand_step = send\<langle>x1\<rangle> \<longrightarrow> P (f1 x1)) \<and> (\<forall>x2. stateful_strand_step = receive\<langle>x2\<rangle> \<longrightarrow> P (f2 x2)) \<and> (\<forall>x31 x32 x33. stateful_strand_step = \<langle>x31: x32 \<doteq> x33\<rangle> \<longrightarrow> P (f3 x31 x32 x33)) \<and> (\<forall>x41 x42. stateful_strand_step = insert\<langle>x41,x42\<rangle> \<longrightarrow> P (f4 x41 x42)) \<and> (\<forall>x51 x52. stateful_strand_step = delete\<langle>x51,x52\<rangle> \<longrightarrow> P (f5 x51 x52)) \<and> (\<forall>x61 x62 x63. stateful_strand_step = \<langle>x61: x62 \<in> x63\<rangle> \<longrightarrow> P (f6 x61 x62 x63)) \<and> (\<forall>x71 x72 x73. stateful_strand_step = \<forall>x71\<langle>\<or>\<noteq>: x72 \<or>\<notin>: x73\<rangle> \<longrightarrow> P (f7 x71 x72 x73)))"
bdd_above_bot: fixes A :: "'k multiset set" shows "subset_mset.bdd_below A"
authlife_LB: shows "Suc 0 \<le> authlife"
HyperN_def: shows "HyperN \<equiv> {b \<in> carrier_vec n. \<nexists>B c. B \<subseteq> A \<and> c \<in> {normal_vector B, - normal_vector B} \<and> Suc (card B) = n \<and> lin_indpt B \<and> (\<forall>a\<^sub>i\<in>A. (0::'a) \<le> c \<bullet> a\<^sub>i) \<and> c \<bullet> b < (0::'a)}"
irreducibleD': fixes a :: "'b"   and b :: "'b" assumes "irreducible a"   and "b dvd a" shows "a dvd b \<or> is_unit b"
assms: shows "ok (e\<lparr>a:X\<rparr>\<lparr>b:Y\<rparr>\<lparr>c:Z\<rparr>)"
i_set_mult_mod_gr0_div_not_in_i_set: fixes k :: "nat"   and d :: "nat" assumes "0 < k"   and "0 < d"   and "0 < k mod d" shows "\<exists>I\<in>i_set_mult k. I \<oslash> d \<notin> i_set"