hcju/isabelleps huggingface dataset:
monic_degree_0: fixes p :: "'a poly" assumes "monic p" shows "(degree p = 0) = (p = 1)"
g: fixes x :: "real"   and y :: "real" assumes "x \<le> y" shows "g x \<le> g y"
next_events_snd: fixes x :: "event option"   and xs :: "event option list" shows "(x \<in> next_events (ts_process {[], [Some b]}) xs) = (xs @ [x] \<in> {[], [Some b]})"
G_def: shows "G = sample_uniform (order \<G>) \<bind> (\<lambda>w. return_spmf ((\<^bold>g [^] w, g' [^] w), w))"
prems_13: shows "InvariantVarsQ (getQ state') F0 Vbl"
th3_4: fixes a :: "(real, 'a) vec"   and b :: "(real, 'a) vec"   and c :: "(real, 'a) vec" assumes "B\<^sub>\<real> a b c"   and "B\<^sub>\<real> b a c" shows "a = b"
ListAskip: fixes xs :: "('a \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat)) list"   and l :: "'a \<Rightarrow> nat"   and s :: "char list \<Rightarrow> int" shows "preList xs SKIP l s = postList xs l s"
circ_plus_one: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" shows "x\<^sup>\<odot> = mone \<oplus> x\<^sup>\<odot>"
Some_eq_filter_option_iff: fixes y :: "'c"   and P :: "'c \<Rightarrow> bool"   and x :: "'c option" shows "(Some y = filter_option P x) = (x = Some y \<and> P y)"
WT_map_converter_idD: fixes \<I> :: "('i, 'j) \<I>"   and \<I>' :: "('k, 'l) \<I>"   and f :: "'i \<Rightarrow> 'k"   and g :: "'l \<Rightarrow> 'j" assumes "\<I>, \<I>' \<turnstile>\<^sub>C map_converter id id f g 1\<^sub>C \<surd>" shows "\<I> \<le> map_\<I> f g \<I>'"
leftderives1_def: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and u :: "symbol list"   and v :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>" shows "CFG.leftderives1 \<NN> \<TT> \<RR> u v = (\<exists>x y N \<alpha>. u = x @ [N] @ y \<and> v = x @ \<alpha> @ y \<and> CFG.is_word \<TT> x \<and> CFG.is_sentence \<NN> \<TT> y \<and> (N, \<alpha>) \<in> \<RR>)"
history_consistent_access_last_read: fixes \<theta> :: "nat \<Rightarrow> nat option"   and p :: "'a"   and volatile :: "bool"   and a :: "nat"   and t :: "nat"   and v :: "nat"   and sb :: "'a memref list" assumes "history_consistent \<theta> p (rev (Read\<^sub>s\<^sub>b volatile a t v # sb))" shows "\<theta> t = Some v"
lcos_mult_one: fixes M :: "'a set" assumes "M \<subseteq> carrier G" shows "\<zero> <#\<^bsub>add_monoid G\<^esub> M = M"
smallomegaD_filterlim_at_top_norm: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> \<omega>[F](g)"   and "\<forall>\<^sub>F x in F. g x \<noteq> (0::'b)" shows "LIM x F. norm (f x / g x) :> at_top"
stabilizer_is_subgroup: fixes m :: "'a set" assumes "m \<in> subgroups_of_size (p ^ a)" shows "subgroup (conj.stabilizer m) G"
that: fixes k :: "nat \<Rightarrow> int" assumes "(\<lambda>j. \<theta> j - real_of_int (k j) * (2 * pi)) \<longlonglongrightarrow> \<Theta>" shows "thesis"
vdomain_vrange_is_vempty: fixes a :: "V"   and b :: "V" shows "(\<D>\<^sub>\<circ> (ZFC_in_HOL.set {\<langle>a, b\<rangle>}) = []\<^sub>\<circ>) = (\<R>\<^sub>\<circ> (ZFC_in_HOL.set {\<langle>a, b\<rangle>}) = []\<^sub>\<circ>)"
indRelLTPO_to_TRel_3: fixes P :: "('procS, 'procT) Proc"   and Q :: "('procS, 'procT) Proc"   and TRel :: "('procT \<times> 'procT) set" assumes "P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>LT<TRel> Q" shows "\<forall>TP SQ. TP \<in>T P \<and> SQ \<in>S Q \<longrightarrow> (TP, \<lbrakk>SQ\<rbrakk>) \<in> (TRel \<union> _Collect (T1, T2) (\<exists>S. T1 = \<lbrakk>S\<rbrakk> \<and> T2 = \<lbrakk>S\<rbrakk>))\<^sup>+"
max_x_def: fixes a :: "nat list"   and b :: "nat list"   and y :: "nat list"   and i :: "nat" shows "hlde_ops.max_x a b y i = (if i < length a \<and> hlde_ops.Di a b i y \<noteq> {} then Min (hlde_ops.Di a b i y) else Max (set b))"
extremum_unique: fixes a :: "'b word" shows "(- 1 \<le> a) = (a = - 1)"
MGT_lemma: fixes c :: "com" assumes "\<forall>p. {} |\<turnstile>\<^sub>t {MGT\<^sub>t (CALL p)}" shows "{} |\<turnstile>\<^sub>t {MGT\<^sub>t c}"
monoI: fixes less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and f :: "'b::type \<Rightarrow> 'c::order" assumes "class.order (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool)"   and "\<And>(x::'b::type) y::'b::type. (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) x y \<Longrightarrow> (f::'b::type \<Rightarrow> 'c::order) x \<le> f y" shows "order.mono (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (f::'b::type \<Rightarrow> 'c::order)"
bot_comp: fixes x :: "'a" shows "\<bottom> * x = \<bottom>"
assms_4: shows "g \<star> f\<^sup>* \<cong> r"
Fvars_psubst: fixes \<phi> :: "'fmla"   and txs :: "('trm \<times> 'var) list" assumes "\<phi> \<in> fmla"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm"   and "distinct (map snd txs)" shows "Fvars (psubst \<phi> txs) = Fvars \<phi> - snd ` set txs \<union> \<Union> {if x \<in> Fvars \<phi> then FvarsT t else {} |t x. (t, x) \<in> set txs}"
has_integral_vec1_nth_cbox: fixes f :: "real \<Rightarrow> 'c"   and y :: "'c"   and a :: "real"   and b :: "real" assumes "(f has_integral y) {a..b}" shows "((\<lambda>x. f (x $ 1)) has_integral y) (cbox (vec a) (vec b))"
qp_P'_Q: shows "is_quantum_predicate (P' + Q)"
cas_induce: fixes u :: "'a"   and p :: "'b list"   and v :: "'a"   and S :: "'a set" assumes "GS.cas u p v"   and "set (GS.awalk_verts u p) \<subseteq> S" shows "pre_digraph.cas ((G \<restriction> S) \<restriction> S) u p v"
gen_mHom_eq: fixes N :: "('g, 'b, 'h) Module_scheme"   and H :: "'a set"   and f :: "'a \<Rightarrow> 'g"   and g :: "'a \<Rightarrow> 'g" assumes "R module N"   and "generator R M H"   and "f \<in> mHom R M N"   and "g \<in> mHom R M N"   and "\<forall>h\<in>H. f h = g h" shows "f = g"
SubobjsR_isClass: 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 C :: "char list"   and Cs :: "char list list" assumes "Subobjs\<^sub>R P C Cs" shows "is_class P C"
alw_ev_sdrop: fixes P :: "'b stream \<Rightarrow> bool"   and n :: "nat"   and w :: "'b stream" shows "alw (ev P) (sdrop n w) = alw (ev P) w"
that: fixes e :: "real" assumes "0 < e"   and "ball x e \<inter> path_image p = {}"   and "ball x e \<inter> {y. 0 < (y - a) \<bullet> rot (a - b)} \<subseteq> inside (path_image (p +++ linepath a b)) \<and> ball x e \<inter> {y. (y - a) \<bullet> rot (a - b) < 0} \<subseteq> outside (path_image (p +++ linepath a b)) \<or> ball x e \<inter> {y. (y - a) \<bullet> rot (a - b) < 0} \<subseteq> inside (path_image (p +++ linepath a b)) \<and> ball x e \<inter> {y. 0 < (y - a) \<bullet> rot (a - b)} \<subseteq> outside (path_image (p +++ linepath a b))" shows "thesis"
strict_implies_order: fixes a :: "'x \<Rightarrow>\<^sub>0 nat"   and b :: "'x \<Rightarrow>\<^sub>0 nat" assumes "a \<prec> b" shows "a \<preceq> b"
Next_def: fixes "next" :: "'b \<Rightarrow> 'b"   and arb :: "'b"   and S :: "'b set" assumes "fresh next" shows "fresh.Next next arb S = (if S = {} then arb else next (Max S))"
curryD: fixes f :: "'b \<times> 'c \<Rightarrow> bool"   and a :: "'b"   and b :: "'c" assumes "curry f a b" shows "f (a, b)"
f: fixes x :: "real" assumes "a \<le> x" shows "(F has_real_derivative f x) (at x)"
FFE: fixes G :: "('a, 'b) form list"   and a :: "('a, 'b) form" assumes "G \<turnstile> FF" shows "G \<turnstile> a"
app_strict: fixes x :: "'e" shows "\<bottom> x = \<bottom>"
e2_def: shows "e2 = min (dist a x) (dist b x)"
add_neg_numeral_special_7: shows "(1::'a) + - (1::'a) = (0::'a)"
no_leading_Cons: fixes P :: "'f \<Rightarrow> bool"   and x :: "'f"   and xs :: "'f list" shows "no_leading P (x # xs) = (\<not> P x)"
UN_simps_4: fixes A :: "'d \<Rightarrow> 'c set"   and B :: "'c set"   and C :: "'d set" shows "(\<Union>x\<in>C. A x \<inter> B) = \<Union> (A ` C) \<inter> B"
nonneg_vel: shows "0 \<le> v\<^sub>o"
psi_def: shows "psi \<equiv> \<lambda>i. if i mod 2 = 0 then psi_u (i div 2) else psi_v (i div 2)"
limseq_shift_iff: fixes x :: "nat \<Rightarrow> 'b"   and i :: "nat"   and y :: "'b" shows "(\<lambda>m. x (m + i)) \<longlonglongrightarrow> y = x \<longlonglongrightarrow> y"
reindex_map_correct: shows "inj_on (reindex_map H) (ta_rstates (hta_\<alpha> H))"
mon_conv_rv: fixes u :: "nat \<Rightarrow> 'b \<Rightarrow> real"   and M :: "'b set set \<times> ('b set \<Rightarrow> real)"   and f :: "'b \<Rightarrow> real" assumes "\<And>n. u n \<in> rv M"   and "u\<up>f" shows "f \<in> rv M"
unset_bit_def: fixes n :: "nat"   and a :: "'a" shows "unset_bit n a = a AND NOT (push_bit n (1::'a))"
gt: fixes numnoabs :: "Ferrack.num \<Rightarrow> bool"   and t :: "Ferrack.num"   and bs :: "real list" assumes "numnoabs t" shows "Ifm bs (case rsplit0 t of (x, xa) \<Rightarrow> gt x xa) = Ifm bs (Gt t) \<and> isrlfm (case rsplit0 t of (x, xa) \<Rightarrow> gt x xa)"
w_def: shows "w = replicate (Suc n\<^sub>0) 0 @ [1] \<odot> 0\<^sup>\<infinity>"
poly_eqI: fixes p :: "linear_poly"   and q :: "linear_poly" assumes "\<And>v. Abstract_Linear_Poly.coeff p v = Abstract_Linear_Poly.coeff q v" shows "p = q"
take_prefix: fixes m :: "nat"   and n :: "nat"   and xs :: "'b list" assumes "m \<le> n" shows "take m xs \<le> take n xs"
mkArr_eqI': fixes A :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr set"   and B :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr set"   and F :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr \<Rightarrow> (hf set, hf \<Rightarrow> hf) concrete_category.arr"   and F' :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr \<Rightarrow> (hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "arr (mkArr A B F)"   and "\<And>x. x \<in> A \<Longrightarrow> F x = F' x" shows "mkArr A B F = mkArr A B F'"
mult_pCons_left: fixes a :: "'b"   and p :: "'b poly"   and q :: "'b poly" shows "pCons a p * q = Polynomial.smult a q + pCons (0::'b) (p * q)"
elabE_5: fixes te :: "ExVar \<Rightarrow> Ty option"   and e1 :: "Ex"   and e2 :: "Ex"   and ty :: "Ty" assumes "te \<turnstile> e1 @@ e2 ===> ty" obtains ty1 :: "Ty" where "te \<turnstile> e1 ===> ty1 -> ty" and "te \<turnstile> e2 ===> ty1"
strict_implies_weak: fixes x :: "'a"   and y :: "'a" assumes "x \<sqsubset> y"   and "x \<in> A"   and "y \<in> A" shows "x \<sqsubseteq> y"
prime_dvd_power_nat_iff: fixes p :: "nat"   and n :: "nat"   and x :: "nat" assumes "prime p"   and "0 < n" shows "(p dvd x ^ n) = (p dvd x)"
abs_ord: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "abs_nat m \<le> abs_nat n"
assms_3: shows "x \<le> y \<squnion> z"
punit_component_of_term: shows "punit.component_of_term = (\<lambda>_. ())"
openin_closedin_eq: fixes U :: "'a topology"   and S :: "'a set" shows "openin U S = (S \<subseteq> topspace U \<and> closedin U (topspace U - S))"
that: fixes a :: "'x \<Rightarrow> 'a" assumes "ideal F = ideal (range (\<lambda>x. monomial (1::'a) (monomial 1 x) - monomial (a x) 0))" shows "thesis"
sorted_wrt_Cons: fixes R :: "('b \<times> 'b) set"   and x :: "'b"   and xs :: "'b list" shows "Linorder_Relations.sorted_wrt R (x # xs) = ((\<forall>y\<in>set xs. (x, y) \<in> R) \<and> Linorder_Relations.sorted_wrt R xs)"
antipar_1: shows "trg\<^sub>C g = src\<^sub>C f"
type_weaken': fixes \<Gamma> :: "binding list"   and t :: "trm"   and T :: "POPLmark.type"   and \<Delta> :: "binding list" assumes "\<Gamma> \<turnstile> t : T"   and "\<Delta> @ \<Gamma> \<turnstile>\<^sub>w\<^sub>f" shows "\<Delta> @ \<Gamma> \<turnstile> \<up> \<parallel>\<Delta>\<parallel> 0 t : \<up>\<^sub>\<tau> \<parallel>\<Delta>\<parallel> 0 T"
cases: fixes a :: "'a list" assumes "walk a" obtains "a = []"   | v :: "'a" where "a = [va__]" and "va__ \<in> V"   | v :: "'a" and w :: "'a" and vs :: "'a list" where "a = v # w # vs" and "v \<rightarrow> w" and "walk (w # vs)"
get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_ok: fixes h :: "(_) heap"   and ptr :: "(_) shadow_root_ptr"   and getter :: "('c, 'a, 'b, 'k option) RShadowRoot_scheme \<Rightarrow> 'm" assumes "ShadowRootClass.type_wf h"   and "ptr |\<in>| shadow_root_ptr_kinds h" shows "h \<turnstile> ok get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t ptr getter"
CJNB_qe: fixes qe :: "fm \<Rightarrow> fm"   and vs :: "'b list" assumes "\<forall>bs p. qfree p \<longrightarrow> qfree (qe p) \<and> Ifm vs bs (qe p) = Ifm vs bs (E p)" shows "\<forall>bs p. qfree p \<longrightarrow> qfree (CJNB qe p) \<and> Ifm vs bs (CJNB qe p) = Ifm vs bs (E p)"
if_multithreaded_def: fixes final :: "'c \<Rightarrow> bool"   and r :: "('f, 'd, 'c, 'e, 'g, 'h) semantics" shows "if_multithreaded final r \<equiv> multithreaded final r"
subseqs_subformulas\<^sub>\<mu>_list: fixes X :: "'a ltln set"   and \<phi> :: "'a ltln" shows "(X \<subseteq> subformulas\<^sub>\<mu> \<phi>) = (\<exists>xs. X = set xs \<and> xs \<in> set (subseqs (subformulas\<^sub>\<mu>_list \<phi>)))"
simps_2: fixes v :: "String.literal"   and va :: "trm list" shows "get_and_list (Pred v va) = [Pred v va]"
interleave_elem_induct: fixes w :: "'b list"   and w1 :: "'b list"   and w2 :: "'b list"   and P :: "'b list \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "w \<in> w1 \<otimes> w2"   and "P [] [] []"   and "\<And>e w w1 w2. \<lbrakk>P w w1 w2; w \<in> w1 \<otimes> w2\<rbrakk> \<Longrightarrow> P (e # w) (e # w1) w2"   and "\<And>e w w1 w2. \<lbrakk>P w w1 w2; w \<in> w1 \<otimes> w2\<rbrakk> \<Longrightarrow> P (e # w) w1 (e # w2)" shows "P w w1 w2"
fail_writer_def: fixes fail :: "'c" shows "fail_writer fail = WriterT fail"
rule_states_lhs: fixes r :: "('a, 'b) ta_rule" shows "lhs r \<in> rule_states r"
induct: fixes P :: "('fun, 'var) Term.term set \<Rightarrow> ('fun, 'var) strand_step list \<Rightarrow> bool"   and a0 :: "('fun, 'var) Term.term set"   and a1 :: "('fun, 'var) strand_step list" assumes "\<And>M. P M []"   and "\<And>M t S. (\<And>x. P M S) \<Longrightarrow> P M (send\<langle>t\<rangle>\<^sub>s\<^sub>t # S)"   and "\<And>M t S. P (insert t M) S \<Longrightarrow> P M (receive\<langle>t\<rangle>\<^sub>s\<^sub>t # S)"   and "\<And>M uu_ t t' S. (\<And>x. P M S) \<Longrightarrow> P M (\<langle>uu_: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S)"   and "\<And>M X F S. (\<And>x. P M S) \<Longrightarrow> P M (\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t # S)" shows "P a0 a1"
1: shows "g \<in> (\<Union>y\<in>S. (\<lambda>g. g(x := y)) ` {g. \<forall>x. g x \<in> f x})"
safe_rpd_fv_regex_le: fixes r :: "'a regex"   and s :: "'a regex"   and test :: "'d \<Rightarrow> 'a \<Rightarrow> bool"   and i :: "'d" assumes "local.safe_regex Past Strict r"   and "s \<in> rpd test i r" shows "fv_regex fv r \<subseteq> fv_regex fv s"
l_1: fixes i :: "'a" assumes "i \<in> I"   and "a < i"   and "i \<le> b" shows "(f \<longlongrightarrow> l i) (at_left i)"
funion_iff: fixes c :: "'a"   and A :: "'a fset"   and B :: "'a fset" shows "(c |\<in>| A |\<union>| B) = (c |\<in>| A \<or> c |\<in>| B)"
assms_1: shows "finite {a. f a \<noteq> (0::'a)}"
intro: fixes G_invar :: "'a \<Rightarrow> bool"   and G_succ :: "'a \<Rightarrow> 'b \<Rightarrow> (nat \<times> 'b) list"   and G_empty :: "'a"   and G_add :: "'b \<times> 'b \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"   and G_\<alpha> :: "'a \<Rightarrow> 'b \<times> 'b \<Rightarrow> enat"   and M_empty :: "'c"   and M_update :: "'b \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> 'c"   and M_delete :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and M_lookup :: "'c \<Rightarrow> 'b \<Rightarrow> nat option"   and M_invar :: "'c \<Rightarrow> bool"   and Q_empty :: "'d"   and Q_update :: "'b \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd"   and Q_delete :: "'b \<Rightarrow> 'd \<Rightarrow> 'd"   and Q_invar :: "'d \<Rightarrow> bool"   and Q_lookup :: "'d \<Rightarrow> 'b \<Rightarrow> nat option"   and Q_is_empty :: "'d \<Rightarrow> bool"   and Q_getmin :: "'d \<Rightarrow> 'b \<times> nat" assumes "adt_finite_wgraph G_invar G_succ G_empty G_add G_\<alpha>"   and "Map M_empty M_update M_delete M_lookup M_invar"   and "PrioMap Q_empty Q_update Q_delete Q_invar Q_lookup Q_is_empty Q_getmin" shows "Dijkstra_Impl_Adts G_\<alpha> G_invar G_succ G_empty G_add M_empty M_update M_delete M_lookup M_invar Q_empty Q_update Q_delete Q_invar Q_lookup Q_is_empty Q_getmin"
1: shows "V.arr \<lparr>Chn = \<mu>.dsrc, Dom = \<lparr>Leg0 = \<mu>.dsrc, Leg1 = \<mu>.dsrc\<rparr>, Cod = \<lparr>Leg0 = \<mu>.dsrc, Leg1 = \<mu>.dsrc\<rparr>\<rparr>"
elims: fixes x :: "Body"   and xa :: "'a Point"   and y :: "'a Cone" assumes "lightcone x xa = y" obtains m :: "Body" and v :: "'a Point" where "x = m" and "xa = v" and "y = mkCone v (c m)"
Constrains_weaken: fixes F :: "'c program"   and A :: "'c set"   and A' :: "'c set"   and B :: "'c set"   and B' :: "'c set" assumes "F \<in> A Co A'"   and "B \<subseteq> A"   and "A' \<subseteq> B'" shows "F \<in> B Co B'"
preserves_reflects_arr: fixes f :: "'a" shows "D.arr (F f) = C.arr f"
inv_inv: fixes f :: "'b" assumes "\<phi>.B.iso f" shows "\<phi>.B.inv (\<phi>.B.inv f) = f"
ccInf_greatest: fixes A :: "'a set"   and z :: "'a" assumes "countable A"   and "\<And>x. x \<in> A \<Longrightarrow> z \<le> x" shows "z \<le> Inf A"
combine_options_simps_2: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b option" shows "combine_options f x None = x"
g_orbital_eq: fixes f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and G :: "'b \<Rightarrow> bool"   and U :: "'b \<Rightarrow> real set"   and S :: "'b set"   and t\<^sub>0 :: "real"   and s :: "'b" shows "x\<acute>=f & G on U S @ t\<^sub>0 s = {X t |t X. t \<in> U s \<and> \<P> X (down (U s) t) \<subseteq> {s. G s} \<and> X \<in> Sols f U S t\<^sub>0 s}"
m_assoc: fixes x :: "'a list"   and y :: "'a list"   and z :: "'a list" assumes "x \<in> carrier (K [X])"   and "y \<in> carrier (K [X])"   and "z \<in> carrier (K [X])" shows "x \<oplus>\<^bsub>K [X]\<^esub> y \<oplus>\<^bsub>K [X]\<^esub> z = x \<oplus>\<^bsub>K [X]\<^esub> (y \<oplus>\<^bsub>K [X]\<^esub> z)"
f: fixes i :: "'i" assumes "i \<in> set_pmf M" shows "continuous_on A (f i)"
jvm_one_step1: fixes \<sigma> :: "jvm_state"   and \<sigma>' :: "jvm_state"   and P :: "jvm_prog"   and \<sigma>'' :: "jvm_state" assumes "P \<turnstile> \<sigma> -jvm\<rightarrow>\<^sub>1 \<sigma>'"   and "P \<turnstile> \<sigma>' -jvm\<rightarrow> \<sigma>''" shows "P \<turnstile> \<sigma> -jvm\<rightarrow> \<sigma>''"
inter_guards_While: fixes cnd :: "'d set"   and bdy1 :: "('d, 'e, 'f) com"   and c2 :: "('d, 'e, 'f) com"   and c :: "('d, 'e, 'f) com" shows "((While cnd bdy1 \<inter>\<^sub>g c2) = Some c) = (\<exists>bdy2 bdy. c2 = While cnd bdy2 \<and> (bdy1 \<inter>\<^sub>g bdy2) = Some bdy \<and> c = While cnd bdy)"
double_compl: fixes x :: "'c" shows "- (- x) = x"
pred_eq_const1: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and N :: "'b measure"   and c :: "'b" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "c \<in> space N \<Longrightarrow> {c} \<in> sets N" shows "Measurable.pred M (\<lambda>x. f x = c)"
sup_extended_case: fixes x1a :: "'a extended"   and x2a :: "'a extended" shows "sup x1a x2a = (case x1a of Fin xda \<Rightarrow> case x2a of Fin x2da \<Rightarrow> Fin (sup xda x2da) | \<infinity> \<Rightarrow> \<infinity> | -\<infinity> \<Rightarrow> Fin xda | \<infinity> \<Rightarrow> case x2a of Fin xba \<Rightarrow> \<infinity> | _ \<Rightarrow> \<infinity> | -\<infinity> \<Rightarrow> case x2a of Fin xca \<Rightarrow> Fin xca | \<infinity> \<Rightarrow> \<infinity> | -\<infinity> \<Rightarrow> -\<infinity>)"
AbGroup: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set" assumes "IrrFinGroupRepresentation G smult V" shows "AbGroup V"
isomorphic_group_triviality1: fixes G :: "('a, 'b) monoid_scheme"   and H :: "('c, 'd) monoid_scheme" assumes "G \<cong> H"   and "Group.group H"   and "trivial_group G" shows "trivial_group H"
vec_nth_inverse: fixes x :: "('a, 'b) Finite_Cartesian_Product.vec" shows "vec_lambda (($h) x) = x"
simpler_gets_def: fixes f :: "'b \<Rightarrow> 'c" shows "gets f = (\<lambda>s. ((f s, s), False))"
card_inj_on_le: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set" assumes "inj_on f A"   and "f ` A \<subseteq> B"   and "finite B" shows "card A \<le> card B"
set_induct: fixes x :: "'b"   and a :: "'b tree"   and P :: "'b \<Rightarrow> 'b tree \<Rightarrow> bool" assumes "x \<in> set_tree a"   and "\<And>z1 z2 z3. P z1 (Node z1 z2 z3)"   and "\<And>z1 z2 z3 xa. \<lbrakk>xa \<in> set_tree z2; P xa z2\<rbrakk> \<Longrightarrow> P xa (Node z1 z2 z3)"   and "\<And>z1 z2 z3 xb. \<lbrakk>xb \<in> set_tree z3; P xb z3\<rbrakk> \<Longrightarrow> P xb (Node z1 z2 z3)" shows "P x a"
less_trm_extension: shows "{(s, t). less_kbo s t} \<subseteq> leq_term"
assms_2: fixes l :: "'a list"   and l' :: "'b list" assumes "a = Array l"   and "a' = Array l'"   and "(l, l') \<in> \<langle>Ra\<rangle>list_rel" shows "(f l, f' l') \<in> Rb"
LIMSEQ_ignore_initial_segment: fixes f :: "nat \<Rightarrow> 'e"   and a :: "'e"   and k :: "nat" assumes "f \<longlonglongrightarrow> a" shows "(\<lambda>n. f (n + k)) \<longlonglongrightarrow> a"
not_less_bot: fixes a :: "'a" shows "\<not> a < \<bottom>"
finite_divisors_nat': fixes n :: "nat" assumes "0 < n" shows "finite {(a, b). a * b = n}"
assms_2: shows "f = g \<or> g \<in> set (P f)"
inject_3: fixes x3 :: "'a \<times> nat"   and y3 :: "'a \<times> nat" shows "(Formal_out x3 = Formal_out y3) = (x3 = y3)"
ac_Diff_iff: fixes c :: "'b"   and A :: "'b antichain"   and B :: "'b antichain" shows "(c \<in>\<^sub>A A - B) = (c \<in>\<^sub>A A \<and> c \<notin>\<^sub>A B)"
\<zeta>: shows "\<zeta> \<notin> path_image p"
value_inv_6: fixes v :: "Syntax.term" shows "\<not> value (Unroll v)"
P2_c: fixes x :: "'a"   and y :: "'a" assumes "x l\<rightarrow> y = (1::'a)"   and "y r\<rightarrow> x = (1::'a)" shows "x = y"
ends_in_V: shows "{u, v, w, x} \<subseteq> V"
lex_ord_pair_ko_ntm_def: fixes ko :: "'a nat_term_order" shows "lex_ord_pair_ko_ntm ko \<equiv> comparator.lex_ord_pair (nat_term_compare_inv ko)"
assms_1: shows "h \<turnstile> get_ancestors ptr \<rightarrow>\<^sub>r ancestors"
Abs_Int_def: fixes num' :: "int \<Rightarrow> 'a"   and plus' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and \<gamma> :: "'a \<Rightarrow> int set" shows "Abs_Int num' plus' \<gamma> \<equiv> Gamma_semilattice num' plus' \<gamma>"
prj_simps_13: shows "C.dom trg_trg_f.Prj\<^sub>1 = Chn (trg f \<star> trg f \<star> f)"
unat_ucast_less_no_overflow_simp: fixes n :: "nat"   and f :: "'b word" assumes "n < 2 ^ LENGTH('b)" shows "(unat f < n) = (f < word_of_nat n)"
discI_2: fixes "term" :: "('c, 'd) Term.term"   and x21 :: "'c"   and x22 :: "('c, 'd) Term.term list" assumes "term = Fun x21 x22" shows "is_Fun term"
MASTER_BOUND_def: fixes a :: "real"   and b :: "real"   and c :: "real"   and x :: "nat" shows "MASTER_BOUND a b c x = real x powr a * ln (real x) powr b * ln (ln (real x)) powr c"
strand_sem_split_1: fixes M :: "('fun, 'var) Term.term set"   and S :: "('fun, 'var) strand_step list"   and S' :: "('fun, 'var) strand_step list"   and \<theta> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "\<lbrakk>M; S @ S'\<rbrakk>\<^sub>c \<theta>" shows "\<lbrakk>M; S\<rbrakk>\<^sub>c \<theta>"
adv: assumes "\<And>\<eta>. exception_\<I> (\<I>_real \<eta> \<oplus>\<^sub>\<I> \<I>_common \<eta>) \<turnstile>g obsf_distinguisher (\<A> \<eta>) \<surd>" shows "Negligible.negligible (\<lambda>\<eta>. advantage (obsf_distinguisher (\<A> \<eta>)) (obsf_resource (sim \<eta> |\<^sub>= 1\<^sub>C \<rhd> ideal_resource \<eta>)) (obsf_resource (real_resource \<eta>)))"
finite_PUpdate: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and S :: "('d, 'e, 'f) status" assumes "TS \<in> HPT S" shows "finite (Expr.PUpdate (Label TS))"
linkageD: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a" assumes "linkage \<Gamma> f"   and "x \<in> A \<Gamma>" shows "d_OUT f x = weight \<Gamma> x"