hcju/isabelleps huggingface dataset:
rel_funD2: fixes A :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and B :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'g"   and g :: "'f \<Rightarrow> 'h"   and x :: "'f" assumes "rel_fun A B f g"   and "A x x" shows "B (f x) (g x)"
simps_5: fixes S :: "'a ltl set" shows "eval\<^sub>G S false = false"
simps_2: fixes a1 :: "'a aexp"   and a2 :: "'a aexp"   and a :: "'a aexp" shows "gexp_constrains (Eq a1 a2) a = (aexp_constrains a1 a \<or> aexp_constrains a2 a)"
abelian_subgroupI2: fixes G :: "('c, 'd) ring_scheme"   and H :: "'c set" assumes "comm_group (add_monoid G)"   and "subgroup H (add_monoid G)" shows "abelian_subgroup H G"
path_split_3: fixes n :: "'b LDCFG_node"   and as :: "('b LDCFG_node \<times> 'c edge_kind \<times> 'b LDCFG_node) list"   and a :: "'b LDCFG_node \<times> 'c edge_kind \<times> 'b LDCFG_node"   and as' :: "('b LDCFG_node \<times> 'c edge_kind \<times> 'b LDCFG_node) list"   and n' :: "'b LDCFG_node" assumes "CFGExit_wf.path n (as @ a # as') n'" shows "CFGExit_wf.path (trg a) as' n'"
Sec: shows "\<forall>l. \<forall>s\<in>Sec - Discl. \<not> \<langle>M l;\<A>;Sec\<rangle> \<turnstile>\<^sub>h\<^sub>o\<^sub>m s"
QuoteP_I: fixes H :: "fm set"   and t :: "tm"   and u :: "tm"   and s :: "tm"   and k :: "tm" assumes "H \<turnstile> SeqQuoteP t u s k" shows "H \<turnstile> QuoteP t u"
bigo_refl: fixes f :: "'c \<Rightarrow> 'd" shows "f \<in> O(f)"
hyps: fixes m :: "nat" assumes "2 ^ n * q < 2 ^ m" shows "q < 2 ^ (m - n)"
is_regular_sequence_def: fixes fs :: "('c \<Rightarrow>\<^sub>0 'd) list" shows "is_regular_sequence fs = (\<forall>j<length fs. \<forall>q. q * fs ! j \<in> ideal (set (take j fs)) \<longrightarrow> q \<in> ideal (set (take j fs)))"
dual_map_linear: fixes f :: "(('a \<Rightarrow> real) \<Rightarrow> real) \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> real" shows "linear_on (dual_space (dest.tangent_space (f p))) (dual_space (src.tangent_space p)) (*\<^sub>R) (*\<^sub>R) (a.dual_map f)"
open_contains_cball: fixes S :: "'b set" shows "open S = (\<forall>x\<in>S. \<exists>e>0. cball x e \<subseteq> S)"
one_eq_divide_iff: 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 a :: "'b"   and b :: "'b" assumes "class.field divide inverse times one plus zero minus uminus" shows "(one = divide a b) = (b \<noteq> zero \<and> a = b)"
lincomb_union2: fixes A :: "'a vec set"   and B :: "'a vec set"   and a :: "'a vec \<Rightarrow> 'a"   and b :: "'a vec \<Rightarrow> 'a" assumes "finite (A \<union> B)"   and "A \<union> B \<subseteq> carrier_vec n"   and "A \<inter> B = {}" shows "lincomb a A + lincomb b B = lincomb (\<lambda>v. if v \<in> A then a v else b v) (A \<union> B)"
entails_def: fixes Bot :: "'a set"   and Q :: "'b set"   and entails_q :: "'b \<Rightarrow> 'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and N1 :: "'a set"   and N2 :: "'a set" assumes "consequence_relation_family Bot Q entails_q" shows "consequence_relation_family.entails Q entails_q N1 N2 = (\<forall>q\<in>Q. entails_q q N1 N2)"
a_plus_one_zero: shows "a + (1::'a) = (0::'a)"
isTrue_exi: fixes x :: "name"   and \<phi> :: "fm" assumes "x \<in> UNIV"   and "\<phi> \<in> UNIV"   and "Fvars \<phi> = {x}"   and "eval_fm e0 (SyntaxN.Ex x \<phi>)" shows "\<exists>n\<in>{t. ground t}. eval_fm e0 (\<phi>(x::=n))"
graph_plan_neq_mems_state_set_neq_len: fixes PROB :: "(('d, 'e) fmap \<times> ('d, 'f) fmap) set"   and vs :: "'d set" shows "prob_dom (prob_proj PROB vs) = prob_dom PROB \<inter> vs"
foundation9: fixes \<tau> :: "'d state \<times> 'd state"   and x :: "'d state \<times> 'd state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "\<tau> \<Turnstile> \<delta> x" shows "(\<tau> \<Turnstile> not x) = (\<tau> |\<noteq> x)"
succ_it_correct: fixes g :: "'G"   and v :: "'V" assumes "invar g" shows "set_iterator (succ_it g v) (succ (\<alpha> g) v)"
D_countable: shows "countable {\<omega> \<in> {0<..<1}. \<not> isCont M.I \<omega>}"
chine_composite: shows "\<nu>\<pi>.chine = \<nu>.leg0 \<down>\<down> \<pi>.leg1"
sat_precond_as_proj: fixes s :: "('d, 'e) fmap"   and as :: "(('d, 'e) fmap \<times> ('d, 'e) fmap) list"   and vs :: "'d set"   and s' :: "('d, 'e) fmap" assumes "sat_precond_as s as"   and "fmrestrict_set vs s = fmrestrict_set vs s'" shows "sat_precond_as s' (as_proj as vs)"
in_TER_plus_current: fixes x :: "'v" assumes "x \<notin> RF\<^sup>\<circ> (TER f)"   and "x \<in> TER\<^bsub>quotient_web \<Gamma> f\<^esub> g" shows "x \<in> TER (plus_current f g)"
of_nat_unat: fixes w :: "'b word" shows "of_nat (unat w) = unsigned w"
kcomp_klift: fixes f :: "'a \<Rightarrow> 'c set"   and g :: "'c \<Rightarrow> 'b set" shows "f \<circ>\<^sub>K g = bd\<^sub>\<F> g \<circ> f"
nn_set_integral_space: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> ennreal" shows "set_nn_integral M (space M) f = integral\<^sup>N M f"
span_directional_derivative: fixes a :: "'b" shows "span (directional_derivative \<infinity> a ` Basis) = manifold_eucl.tangent_space \<infinity> a"
R: fixes si :: "'a" assumes "wf R"   and "P \<Longrightarrow>\<^sub>A \<up> (s = si) * I s * true"   and "\<And>si s. <\<up> (s = si) * I s * true> bi si <\<lambda>r. \<up> (s = si) * I s * true * \<up> (r = b s)>\<^sub>t"   and "\<And>si s. b s \<Longrightarrow> <\<up> (s = si) * I s * true> f si <\<lambda>r. \<exists>\<^sub>As'. \<up> (s' = r) * I s' * true * \<up> ((s', s) \<in> R)>\<^sub>t"   and "\<And>si s. \<not> b s \<Longrightarrow> \<up> (s = si) * I s * true \<Longrightarrow>\<^sub>A \<up> (s = si) * Q s * true" shows "<P> heap_WHILET bi f si <\<lambda>r. \<exists>\<^sub>As. \<up> (s = r) * Q s * true>\<^sub>t"
npClose: fixes R :: "('d, 'e) Ring_scheme"   and x :: "'d"   and n :: "nat" assumes "Ring R"   and "x \<in> carrier R" shows "x^\<^bsup>R n\<^esup> \<in> carrier R"
gcd_dvd2: fixes a :: "'a"   and b :: "'a" shows "normalization_euclidean_semiring_class.gcd a b dvd b"
orelse_eq_FF_iff: fixes x :: "tr"   and y :: "tr" shows "((x orelse y) = FF) = (x = FF \<and> y = FF)"
rep_eq: fixes x :: "('c, 'd) fmap" shows "fset (fmdom x) = dom (($$) x)"
any_\<sigma>: fixes bs :: "bool list"   and n :: "nat" assumes "length bs = n" shows "(any, bs) \<in> set (\<sigma> \<Sigma> n)"
rcoset_mult_add: fixes I :: "'e set"   and R :: "('e, 'f) ring_scheme"   and x :: "'e"   and y :: "'e" assumes "ideal I R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "[mod I:] (I +>\<^bsub>R\<^esub> x) \<Otimes>\<^bsub>R\<^esub> (I +>\<^bsub>R\<^esub> y) = I +>\<^bsub>R\<^esub> x \<otimes>\<^bsub>R\<^esub> y"
read_only_unowned_instructions_read_value_independent: fixes i :: "nat"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and p' :: "'a"   and is' :: "instr list"   and \<theta>' :: "nat \<Rightarrow> nat option"   and \<D>' :: "bool"   and \<R>' :: "nat \<Rightarrow> bool option" assumes "i < length ts\<^sub>s\<^sub>b"   and "ts\<^sub>s\<^sub>b ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "read_only_reads_unowned (ts\<^sub>s\<^sub>b[i := (p', is', \<theta>', sb, \<D>', \<O>, \<R>')])"
ord_eq_le_trans: fixes a :: "'d"   and b :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and c :: "'d" assumes "a = b"   and "less_eq b c" shows "less_eq a c"
prems_4: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (trms\<^sub>s\<^sub>t (S @ \<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S'))"
prec_first_EV_SIGNAL_action: fixes aseq :: "int_point_t list"   and s :: "rstate_t"   and d :: "thread_id_t" assumes "is_SK_EV_SIGNAL aseq" shows "rprecondition s d (hd aseq)"
complex_root_unity: fixes n :: "nat"   and j :: "nat" assumes "n \<noteq> 0" shows "exp (2 * complex_of_real pi * \<i> * of_nat j / of_nat n) ^ n = 1"
InterD: fixes A :: "'c"   and C :: "'c set set"   and X :: "'c set" assumes "A \<in> \<Inter> C"   and "X \<in> C" shows "A \<in> X"
fbd_iso: fixes X :: "'c set"   and Y :: "'c set"   and f :: "'c \<Rightarrow> 'd set" shows "X \<subseteq> Y \<longrightarrow> bd\<^sub>\<F> f X \<subseteq> bd\<^sub>\<F> f Y"
llist_map_def: shows "llist_map \<equiv> \<Lambda> a. \<mu> uu. llist_abs oo ssum_map\<cdot>ID\<cdot>(sprod_map\<cdot>(u_map\<cdot>a)\<cdot>(u_map\<cdot>uu)) oo llist_rep"
WBT_def: fixes \<Delta>1 :: "nat"   and \<Delta>2 :: "nat"   and \<Gamma>1 :: "nat"   and \<Gamma>2 :: "nat" shows "WBT \<Delta>1 \<Delta>2 \<Gamma>1 \<Gamma>2 \<equiv> \<Delta>2 < \<Delta>1 \<and> \<Gamma>2 < \<Gamma>1"
comparator_of_def: fixes x :: "(nat, nat) pp"   and y :: "(nat, nat) pp" shows "lex.comparator_of x y = (if lex.lt x y then Lt else if x = y then Eq else Gt)"
distinctPermCancel: fixes p :: "(name \<times> name) list"   and T :: "'c" assumes "distinctPerm p" shows "p \<bullet> p \<bullet> T = T"
cont_eff_ParT_BrnFT_L: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('test, 'atom, 'choice) cmd list"   and ii :: "nat"   and P :: "nat \<Rightarrow> nat set set"   and F :: "nat \<Rightarrow> nat set \<Rightarrow> nat set"   and s :: "'state"   and t :: "'state" assumes "length cl = length dl"   and "properL cl"   and "properL dl"   and "SbisL cl dl"   and "ii \<in> BrnFT cl dl"   and "\<And>n I i j. \<lbrakk>n < length cl; I \<in> P n; i \<in> I; j \<in> F n I\<rbrakk> \<Longrightarrow> eff (cl ! n) s i \<approx> eff (dl ! n) t j \<and> local.cont (cl ! n) s i \<approx>s local.cont (dl ! n) t j" shows "s \<approx> eff (ParT cl) s ii \<and> (local.cont (ParT cl) s ii, ParT dl) \<in> thetaZOParT"
assms_2: fixes v :: "'a" assumes "v \<in> A"   and "v \<in> VV p"   and "\<not> deadend v" shows "v \<rightarrow> \<sigma>' v"
span_scale: fixes x :: "('a, 'b) vec"   and S :: "('a, 'b) vec set"   and c :: "'a" assumes "x \<in> vec.span S" shows "c *s x \<in> vec.span S"
edges_def: shows "edges \<equiv> \<lambda>(l, g, p). {(l, g, p, X, l') |X l'. (X, l') \<in> set_pmf p}"
distinct_29: fixes x31 :: "poscheckvariant"   and x32 :: "('c, 'd) Term.term"   and x33 :: "('c, 'd) Term.term"   and x71 :: "'d list"   and x72 :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list"   and x73 :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list" shows "\<langle>x31: x32 \<doteq> x33\<rangle> \<noteq> \<forall>x71\<langle>\<or>\<noteq>: x72 \<or>\<notin>: x73\<rangle>"
add_cancel_right_left: fixes a :: "'ag"   and b :: "'ag" assumes "a \<in> U"   and "b \<in> U" shows "(a = b +\<^sub>o\<^sub>w a) = (b = 0\<^sub>o\<^sub>w)"
L1a_Converter: fixes z :: "nat \<Rightarrow> bit list"   and x :: "nat \<Rightarrow> bit list"   and t :: "nat" assumes "Converter z x"   and "fin_make_untimed (inf_truncate z t) \<noteq> []"   and "hd (x t) = Zero" shows "fin_make_untimed (inf_truncate z t) ! (length (fin_make_untimed (inf_truncate z t)) - 1) = Zero"
BIT_config'_fin: fixes s :: "('a list \<times> bool list \<times> 'a list) pmf"   and qs :: "'a list" assumes "finite (set_pmf s)" shows "finite (set_pmf (Partial_Cost_Model.config'_rand BIT s qs))"
bunch_of_lemmata_about_matches_2: fixes \<gamma> :: "('a \<Rightarrow> 'b \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'b \<Rightarrow> bool)"   and a :: "action"   and p :: "'b" shows "matches \<gamma> MatchAny a p"
llength_simps_2: fixes r :: "'b llist"   and A :: "'b set"   and a :: "'b" assumes "r \<in> A\<^sup>\<star>" shows "LList2.llength (a ## r) = Suc (LList2.llength r)"
FORK\<^sub>n\<^sub>o\<^sub>r\<^sub>m_def: fixes i :: "nat" shows "FORK\<^sub>n\<^sub>o\<^sub>r\<^sub>m i = P\<^sub>n\<^sub>o\<^sub>r\<^sub>m\<lbrakk>Tr\<^sub>f i,Up\<^sub>f i\<rbrakk>"
prod_4: fixes b :: "'a N"   and a :: "'a N" shows "b r\<rightarrow> b * b \<le> a \<sqinter> (a r\<rightarrow> b) r\<rightarrow> b"
l_one: fixes x :: "Vertex \<Rightarrow> Vertex" assumes "x \<in> carrier tetrahedral_group" shows "\<one>\<^bsub>tetrahedral_group\<^esub> \<otimes>\<^bsub>tetrahedral_group\<^esub> x = x"
"3_2": fixes t :: "('d, 'c) Term.term"   and \<theta> :: "'c \<Rightarrow> ('d, 'c) Term.term" shows "fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p (receive\<langle>t\<rangle> \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta>) = fv (t \<cdot> \<theta>)"
that: fixes r :: "'a poly"   and s :: "'a poly" assumes "0 < degree r"   and "0 < degree s"   and "coprime r s"   and "p = r * s" shows "thesis"
simplex_furthest_le: fixes S :: "'b set"   and a :: "'b" assumes "finite S"   and "S \<noteq> {}" shows "\<exists>y\<in>S. \<forall>x\<in>convex hull S. norm (x - a) \<le> norm (y - a)"
pwd_spy_con_rel: fixes n :: "nat" assumes "n \<in> bad_prikey \<inter> bad_id_shakey" shows "s\<^sub>0 \<Turnstile> fst (pwd_spy_con n)"
order_less_asym: fixes x :: "'b"   and y :: "'b"   and P :: "bool" assumes "x < y"   and "\<not> P \<Longrightarrow> y < x" shows "P"
is_natural_transformation: shows "natural_transformation A B F F \<sigma>\<sigma>'.map"
Derives1: shows "LeftDerives1 (u @ v) (fst d + length u) (snd d) (u @ x)"
zimage_id: fixes A :: "'a zet" shows "zimage id A = A"
AllForm: shows "\<forall>nP G. sinvar G nP = (\<forall>(e1, e2)\<in>edges G. P e1 e2 nP G)"
findim_domain_findim_image: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and smult' :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "'b \<Rightarrow> 'c" assumes "VectorSpaceHom smult V smult' T"   and "\<exists>ms. set ms \<subseteq> V \<and> R_scalar_mult.RSpan UNIV smult ms = V" shows "\<exists>ms. set ms \<subseteq> T ` V \<and> R_scalar_mult.RSpan UNIV smult' ms = T ` V"
subseq_Cons2': fixes x :: "'a"   and xs :: "'a list"   and ys :: "'a list" assumes "subseq (x # xs) (x # ys)" shows "subseq xs ys"
real_arch_pow: fixes x :: "real"   and y :: "real" assumes "1 < x" shows "\<exists>n. y < x ^ n"
integrable_neg: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set" assumes "f integrable_on S" shows "(\<lambda>x. - f x) integrable_on S"
swap_simps_2: fixes a :: "NominalSequents.var"   and b :: "NominalSequents.var" shows "[(a, b)] \<bullet> b = a"
unfold: shows "eq = (\<Lambda> va v. Fixrec.run\<cdot>(match_snil\<cdot>va\<cdot>(match_snil\<cdot>v\<cdot>(Fixrec.succeed\<cdot>TT)) +++ match_scons\<cdot>va\<cdot>(\<Lambda> x xs. match_snil\<cdot>v\<cdot>(Fixrec.succeed\<cdot>FF)) +++ match_snil\<cdot>va\<cdot>(match_scons\<cdot>v\<cdot>(\<Lambda> y ys. Fixrec.succeed\<cdot>FF)) +++ match_scons\<cdot>va\<cdot>(\<Lambda> x xs. match_scons\<cdot>v\<cdot>(\<Lambda> y ys. Fixrec.succeed\<cdot>(eq\<cdot>x\<cdot>y andalso eq\<cdot>xs\<cdot>ys)))))"
rep_eq: fixes x :: "'a ltl_prop_equiv_quotient"   and xa :: "'a ltl_prop_equiv_quotient" shows "local.ltl_prop_equiv_quotient_eq_test x xa = Rep x \<equiv>\<^sub>P Rep xa"
genlist_simps_1: fixes f :: "nat \<Rightarrow> 'a" shows "genlist f 0 = []"
maddux_20: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'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 bot :: "'b"   and top :: "'b"   and composition :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and converse :: "'b \<Rightarrow> 'b"   and unit :: "'b"   and x :: "'b" assumes "class.relation_algebra minus uminus inf less_eq less sup bot top composition converse unit" shows "less_eq x (composition x top)"
holomorphic_on_inverse: fixes f :: "complex \<Rightarrow> complex"   and s :: "complex set" assumes "f holomorphic_on s"   and "\<And>z. z \<in> s \<Longrightarrow> f z \<noteq> 0" shows "(\<lambda>z. inverse (f z)) holomorphic_on s"
drop_suffix_hd_css_step': fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and pcs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "('e, 'f) xstate"   and cs' :: "('e, 'd, 'f) com list"   and css'css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and t :: "('e, 'f) xstate"   and p :: "('e, 'd, 'f) com"   and ps :: "('e, 'd, 'f) com list"   and cs :: "('e, 'd, 'f) com list"   and pnorm :: "('e, 'd, 'f) com list"   and pabr :: "('e, 'd, 'f) com list" assumes "\<Gamma>\<turnstile> (pcs, css, s) \<rightarrow> (cs', css'css, t)"   and "pcs = p # ps @ cs"   and "css'css = (pnorm @ cs, pabr @ cs) # css" shows "\<Gamma>\<turnstile> (p # ps, css, s) \<rightarrow> (cs', (pnorm, pabr) # css, t)"
det30_zero_scaleR1: fixes e :: "real"   and xr :: "real \<times> real"   and P :: "real \<times> real" assumes "0 < e"   and "coll 0 xr P" shows "coll 0 (e *\<^sub>R xr) P"
swap_def: fixes x :: "'b \<times> 'a" shows "swap x \<equiv> (snd x, fst x)"
impI: fixes cid :: "'kid"   and t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'KS"   and n :: "nat"   and \<gamma> :: "(nat \<Rightarrow> 'KS) \<Rightarrow> nat \<Rightarrow> bool"   and \<gamma>' :: "(nat \<Rightarrow> 'KS) \<Rightarrow> nat \<Rightarrow> bool" assumes "sb.eval cid t t' n \<gamma> \<longrightarrow> sb.eval cid t t' n \<gamma>'" shows "sb.eval cid t t' n (\<gamma> \<longrightarrow>\<^sup>b \<gamma>')"
assms: shows "x \<in>\<^sub>\<circ> VLambda A f"
rel_carrier_Eps_in_3: fixes X :: "('a set \<times> 'b) set" assumes "X \<in> carrier_direct_lim" shows "X = \<lfloor>fst (SOME x. x \<in> X), snd (SOME x. x \<in> X)\<rfloor>"
value_written_committed_def: fixes P :: "'a prog"   and E :: "('b \<times> ('c, 'b) obs_event action) llist"   and J :: "nat \<Rightarrow> ('c, 'b) justifying_execution" shows "value_written_committed P E J = (\<forall>n. \<forall>w\<in>write_actions (justifying_exec (J n)) \<inter> committed (J n). let w' = action_translation (J n) w in \<forall>adal\<in>action_loc P E w'. value_written P (justifying_exec (J n)) w adal = value_written P E w' adal)"
relcompI: fixes a :: "'d"   and b :: "'e"   and r :: "('d \<times> 'e) set"   and c :: "'f"   and s :: "('e \<times> 'f) set" assumes "(a, b) \<in> r"   and "(b, c) \<in> s" shows "(a, c) \<in> r O s"
effect_guardI: fixes P :: "heap \<Rightarrow> bool"   and h :: "heap"   and h' :: "heap"   and f :: "heap \<Rightarrow> 'a \<times> heap"   and r :: "'a" assumes "P h"   and "h' = snd (f h)"   and "r = fst (f h)" shows "effect (guard P f) h h' r"
NS3_msg_in_parts_spies: fixes S :: "agent"   and A :: "agent"   and KA :: "nat"   and N :: "msg"   and B :: "msg"   and K :: "msg"   and X :: "msg"   and evs :: "event list" assumes "Says S A (Crypt KA \<lbrace>N, B, K, X\<rbrace>) \<in> set evs" shows "X \<in> parts (knows Spy evs)"
lconf_def: fixes P :: "'b prog"   and h :: "'heap"   and l :: "String.literal \<Rightarrow> 'addr val option"   and E :: "String.literal \<Rightarrow> ty option" shows "h.lconf P h l E \<equiv> \<forall>V v. l V = \<lfloor>v\<rfloor> \<longrightarrow> (\<exists>T. E V = \<lfloor>T\<rfloor> \<and> h.conf P h v T)"
people_finite: fixes blue :: "'a"   and brown :: "'a"   and green :: "'a"   and red :: "'a"   and S :: "'b set" assumes "distinct [blue, brown, green, red]"   and "finite UNIV" shows "finite S"
measurable_SUP1: fixes i :: "'b"   and I :: "'b set"   and f :: "'c \<Rightarrow> 'd"   and M :: "'b \<Rightarrow> 'c measure"   and N :: "'d measure" assumes "i \<in> I"   and "f \<in> M i \<rightarrow>\<^sub>M N"   and "\<And>m n. \<lbrakk>m \<in> I; n \<in> I\<rbrakk> \<Longrightarrow> space (M m) = space (M n)" shows "f \<in> \<Squnion> (M ` I) \<rightarrow>\<^sub>M N"
prj_simps_3: shows "C.arr \<mu>_\<nu>_\<pi>\<rho>.Prj\<^sub>0"
comp_in_homI': fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a"   and a :: "'a \<times> 'a"   and c :: "'a \<times> 'a" assumes "CC.arr f"   and "CC.arr g"   and "CC.dom f = a"   and "CC.cod g = c"   and "CC.dom g = CC.cod f" shows "CC.in_hom (CC.comp g f) a c"
firstInNet: fixes a :: "('b, 'c) Combinators"   and p :: "('b, 'c) Combinators list" assumes "\<not> member DenyAll a" shows "first_destNet a \<in> set (net_list_aux (a # p))"
intro: fixes p :: "nat"   and n :: "nat"   and b_s :: "nat \<Rightarrow> nat"   and u_s :: "nat \<Rightarrow> nat"   and s :: "(nat \<Rightarrow> nat) set"   and b_t :: "nat \<Rightarrow> nat"   and u_t :: "nat \<Rightarrow> nat"   and t :: "(nat \<Rightarrow> nat) set" assumes "kuhn_simplex p n b_s u_s s"   and "kuhn_simplex p n b_t u_t t" shows "kuhn_simplex_pair p n b_s u_s s b_t u_t t"
periodic_arg_plus_0: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g1 :: "'b \<Rightarrow> 'b"   and gn1 :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "periodic_fun f g gm g1 gn1" shows "g x (0::'b) = x"
fds_const_one: shows "fds_const (1::'b) = 1"
semigroup_mult_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "semigroup_mult_hom hom \<equiv> times_hom hom"
IH_1: assumes "avl l" shows "avl (AVL.insert x l)"
simps_2: fixes M :: "(('fun, 'c, 'sets) prot_fun, 'd) Term.term set"   and TI :: "('sets set \<times> 'sets set) list"   and f :: "('fun, 'c, 'sets) prot_fun"   and T :: "(('fun, 'c, 'sets) prot_fun, 'd) Term.term list" shows "intruder_synth_mod_eq_timpls M TI (Fun f T) = ((\<exists>t\<in>M. equal_mod_timpls TI t (Fun f T)) \<or> public f \<and> length T = arity f \<and> list_all (intruder_synth_mod_eq_timpls M TI) T)"
increase_i_bound: fixes upw :: "bool"   and i :: "nat"   and fs :: "int vec list" assumes "LLL_bound_invariant True upw i fs"   and "i < m"   and "upw \<Longrightarrow> i = 0"   and "i \<noteq> 0 \<Longrightarrow> \<parallel>gram_schmidt_fs.gso n (map of_int_hom.vec_hom fs) (i - 1)\<parallel>\<^sup>2 \<le> \<alpha> * \<parallel>gram_schmidt_fs.gso n (map of_int_hom.vec_hom fs) i\<parallel>\<^sup>2" shows "LLL_bound_invariant True True (Suc i) fs"
simps: fixes A :: "('c, 'd) nba"   and p :: "'d"   and a :: "'d" shows "transition_system.reachablep (\<lambda>a p. snd a) (\<lambda>a p. fst a \<in> alphabet A \<and> snd a \<in> transition A (fst a) p) p a = (a = p \<or> (\<exists>q aa. a = snd aa \<and> transition_system.reachablep (\<lambda>a p. snd a) (\<lambda>a p. fst a \<in> alphabet A \<and> snd a \<in> transition A (fst a) p) p q \<and> fst aa \<in> alphabet A \<and> snd aa \<in> transition A (fst aa) q))"
Rep_filter_inject: fixes x :: "'b Filters.filter"   and y :: "'b Filters.filter" shows "(Filters.filter.Rep_filter x = Filters.filter.Rep_filter y) = (x = y)"
unat_of_nat: fixes x :: "nat" shows "unat (word_of_nat x) = x mod 2 ^ LENGTH('b)"
klift_prop: fixes f :: "'b \<Rightarrow> 'a set"   and X :: "'b set" shows "(f\<^sup>\<dagger>) X = \<mu> (\<P> f X)"
is_root_def: fixes p :: "'a list list"   and x :: "'a list" shows "UP.is_root p x = (x \<in> carrier (poly_ring R) \<and> UP.eval p x = \<zero>\<^bsub>poly_ring R\<^esub> \<and> p \<noteq> [])"
compl: fixes p :: "('a, 'b) form" shows "compl p = Neg p \<or> (\<exists>q. compl p = q \<and> p = Neg q)"
is_ground_cls_list_def: fixes CC :: "('a, 'b) Term.term literal multiset list" shows "is_ground_cls_list CC = Ball (set CC) is_ground_cls"
is_aligned_neg_mask2: fixes a :: "'a word"   and n :: "nat" shows "is_aligned (a && ~~ mask n) n"
decisive_def: fixes scf :: "('c \<Rightarrow> ('d \<times> 'd) set) \<Rightarrow> ('d \<times> 'd) set"   and A :: "'d set"   and Is :: "'c set"   and C :: "'c set"   and x :: "'d"   and y :: "'d" shows "decisive scf A Is C x y \<equiv> C \<subseteq> Is \<and> (\<forall>P. profile A Is P \<and> (\<forall>i\<in>C. x \<^bsub>(P i)\<^esub>\<prec> y) \<longrightarrow> x \<^bsub>(scf P)\<^esub>\<prec> y)"
image_image_fixed_card_subset: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and k :: "nat" assumes "inj_on f A" shows "(`) f ` {X. X \<subseteq> A \<and> card X = k} = {X. X \<subseteq> f ` A \<and> card X = k}"
tensor_mat_add2: fixes m1 :: "'a mat"   and m2 :: "'a mat"   and m3 :: "'a mat" assumes "m1 \<in> carrier_mat d1 d1"   and "m2 \<in> carrier_mat d2 d2"   and "m3 \<in> carrier_mat d2 d2" shows "tensor_mat m1 (m2 + m3) = tensor_mat m1 m2 + tensor_mat m1 m3"