hcju/isabelleps huggingface dataset:
Dom_in_Obj: fixes f :: "('obj, 'obj) concrete_category.arr" assumes "arr f" shows "Dom f \<in> Obj"
simps: fixes as :: "atom set"   and x :: "'e" shows "supp_set ([as]set. x) = supp x - as"
r_neg1: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "y \<in> carrier Zp" shows "\<ominus> x \<oplus> (x \<oplus> y) = y"
unique_rootD_4: fixes plr :: "int poly \<times> rat \<times> rat" assumes "\<exists>!x. root_cond plr x" shows "root_cond plr (real_of_1 plr)"
subR: shows "P \<turnstile> X \<prec>\<^sub>R Y"
hom: shows "homotopic_with_canon (\<lambda>x. True) T U f g"
USdB_decomp_head_tail: fixes c :: "('exp, 'id) MWLfCom"   and V :: "('exp, 'id) MWLfCom list"   and c' :: "('exp, 'id) MWLfCom"   and V' :: "('exp, 'id) MWLfCom list"   and d :: "'d" assumes "(c # V) \<approx>\<^bsub>d\<^esub> c' # V'" shows "[c] \<approx>\<^bsub>d\<^esub> [c'] \<and> V \<approx>\<^bsub>d\<^esub> V'"
nres_bind_let_law: fixes v :: "'c"   and f :: "'c \<Rightarrow> 'b nres"   and g :: "'b \<Rightarrow> 'a nres" shows "Let v f \<bind> g = (let y = v in f y \<bind> g)"
Q2_a_def: shows "Q2_a \<equiv> [: X, mrk \<leadsto> X', mrk' . root \<noteq> nil \<and> X' = {root} \<and> mrk' = {root} :]"
Says_tgs_message_form: fixes Tgs' :: "agent"   and A :: "agent"   and authK :: "nat"   and servK :: "nat"   and B :: "agent"   and Ts :: "msg"   and servTicket :: "msg"   and evs :: "event list" assumes "Says Tgs' A (Crypt authK \<lbrace>Key servK, Agent B, Ts, servTicket\<rbrace>) \<in> set evs"   and "authK \<in> symKeys"   and "evs \<in> kerbIV" shows "servK \<notin> range shrK \<and> (\<exists>A. servTicket = Crypt (shrK B) \<lbrace>Agent A, Agent B, Key servK, Ts\<rbrace>) \<or> servTicket \<in> analz (knows Spy evs)"
trancl_Int_subset: fixes r :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set" assumes "r \<subseteq> s"   and "(r\<^sup>+ \<inter> s) O r \<subseteq> s" shows "r\<^sup>+ \<subseteq> s"
prj_simps_11: shows "C.seq \<nu>\<pi>_\<rho>.prj\<^sub>1 \<p>\<^sub>0[\<mu>.leg0, \<nu>\<pi>.leg1 \<cdot> \<nu>\<pi>_\<rho>.prj\<^sub>1]"
map_comp: fixes g1 :: "'f \<Rightarrow> 'd"   and g2 :: "'g \<Rightarrow> 'e"   and f1 :: "'h \<Rightarrow> 'f"   and f2 :: "'i \<Rightarrow> 'g"   and v :: "('h, 'i) F" shows "F g1 g2 (F f1 f2 v) = F (g1 \<circ> f1) (g2 \<circ> f2) v"
sel_4: fixes x21 :: "'h"   and x22 :: "'f \<Rightarrow> ('g, 'h, 'f) Resumption.resumption" shows "Resumption.resumption.resume (Resumption.resumption.Pause x21 x22) = x22"
add: fixes b1 :: "'b"   and b2 :: "'b" shows "((- 1) ^ Suc n / fact n) *\<^sub>R (b1 + b2) = ((- 1) ^ Suc n / fact n) *\<^sub>R b1 + ((- 1) ^ Suc n / fact n) *\<^sub>R b2"
simps_2: fixes b :: "'t \<Rightarrow>\<^sub>0 'b"   and bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and p :: "'t \<Rightarrow>\<^sub>0 'b" shows "new_spairs (b # bs) p = (if is_regular_spair p b then insort_wrt pair_ord (Inl (p, b)) (new_spairs bs p) else new_spairs bs p)"
invh_combine: fixes l :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and r :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" assumes "invh l"   and "invh r"   and "bheight l = bheight r" shows "invh (combine l r) \<and> bheight (combine l r) = bheight l"
when_def: fixes zero :: "'b"   and a :: "'b"   and P :: "bool" shows "zero.when zero a P = (if P then a else zero)"
merge_def: fixes ws :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and idx :: "nat \<Rightarrow> nat" shows "merge ws idx \<equiv> \<lambda>n. let i = THE i. n \<in> {idx i..<idx (Suc i)} in ws i n"
InvariantUniqQAfterAssertLiteral: fixes state :: "State"   and literal :: "Literal"   and decision :: "bool" assumes "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantUniqQ (getQ state)" shows "let state' = assertLiteral literal decision state in InvariantUniqQ (getQ state')"
lemma_W1: fixes a :: "'b G"   and b :: "'b G" shows "(a l\<rightarrow> b) r\<rightarrow> b = (b l\<rightarrow> a) r\<rightarrow> a"
nb: fixes z :: "complex" assumes "bounded (connected_component_set (- S) z)" shows "z \<in> S"
iUntil_True: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool" shows "(P t'. t' \<U> t I. True) = (I \<noteq> {})"
proper_prappend_signed_list: fixes xs :: "('b \<times> bool) list"   and ys :: "('b \<times> bool) list" assumes "proper_signed_list xs"   and "proper_signed_list ys" shows "proper_signed_list (prappend_signed_list xs ys)"
iteratei_abs_rule_P: fixes \<alpha> :: "'e \<Rightarrow> 'f"   and invar :: "'e \<Rightarrow> bool"   and iti :: "('g \<Rightarrow> bool) \<Rightarrow> ('e \<Rightarrow> 'g \<Rightarrow> 'g) \<Rightarrow> 'g \<Rightarrow> 'g"   and S0 :: "'f set"   and R :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and I :: "'f set \<Rightarrow> 'g \<Rightarrow> bool"   and \<sigma>0 :: "'g"   and c :: "'g \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'g \<Rightarrow> 'g"   and P :: "'g \<Rightarrow> bool" assumes "set_iterator_abs_genord \<alpha> invar iti S0 R"   and "I S0 \<sigma>0"   and "\<And>S \<sigma> x. \<lbrakk>c \<sigma>; invar x; \<alpha> x \<in> S; I S \<sigma>; S \<subseteq> S0; \<forall>y\<in>S - {\<alpha> x}. R (\<alpha> x) y; \<forall>y\<in>S0 - S. R y (\<alpha> x)\<rbrakk> \<Longrightarrow> I (S - {\<alpha> x}) (f x \<sigma>)"   and "\<And>\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> S. \<lbrakk>S \<subseteq> S0; S \<noteq> {}; \<not> c \<sigma>; I S \<sigma>; \<forall>x\<in>S. \<forall>y\<in>S0 - S. R y x\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (iti c f \<sigma>0)"
reds_avoids_live': fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and L :: "Vars.var list"   and \<Delta> :: "(Vars.var \<times> exp) list"   and z :: "exp" assumes "\<Gamma> : e \<Down>\<^bsub>L\<^esub> \<Delta> : z" shows "(domA \<Delta> - domA \<Gamma>) \<inter> set L = {}"
abs_eq: fixes xa :: "perm"   and x :: "('a, 'b, 'c) Formula.Tree" shows "xa \<bullet> Formula.abs_Tree\<^sub>\<alpha> x = Formula.abs_Tree\<^sub>\<alpha> (xa \<bullet> x)"
th0: shows "a ^ Suc m = (\<Prod>i = 0..m. a)"
a2: shows "r i \<cdot> ad t \<le> r q"
lmeasure_integral_UNIV: fixes S :: "'a set" assumes "S \<in> lmeasurable" shows "Sigma_Algebra.measure lebesgue S = integral UNIV (indicat_real S)"
sel_2: fixes x2 :: "'a" shows "hd.sym (Sym x2) = x2"
nd_gw: shows "\<not> (diverge e \<or> goes_wrong e)"
underS_UnderS_singl: fixes r :: "('a \<times> 'a) set"   and a :: "'a" shows "underS r a = UnderS r {a}"
lsided_prop: fixes x :: "'a"   and y :: "'a" assumes "lsd x" shows "y \<cdot> x \<le> x"
distinct_map_snd_inj: fixes l :: "('b \<times> 'a) list"   and a :: "'b"   and b :: "'a"   and a' :: "'b" assumes "distinct (map snd l)"   and "(a, b) \<in> set l"   and "(a', b) \<in> set l" shows "a = a'"
has_derivative_at_within_mono: fixes t :: "real"   and T :: "real set"   and S :: "real set"   and f :: "real \<Rightarrow> 'b"   and f' :: "real \<Rightarrow> 'b" assumes "t \<in> interior T"   and "T \<subseteq> S"   and "D f \<mapsto> f' at t within T" shows "D f \<mapsto> f' at t within S"
filter_empty_ivls_impl_def: fixes le :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and ivls :: "('c \<times> 'd) list" shows "filter_empty_ivls_impl le ivls = filter (\<lambda>(i, s). le i s) ivls"
ex_assn_def: fixes P :: "'b \<Rightarrow> assn" shows "(\<exists>\<^sub>Ax. P x) = Abs_assn (Assn (\<lambda>h. \<exists>x. h \<Turnstile> P x))"
r_neg1: fixes x :: "'c"   and y :: "'c" assumes "x \<in> carrier V"   and "y \<in> carrier V" shows "\<ominus>\<^bsub>V\<^esub> x \<oplus>\<^bsub>V\<^esub> (x \<oplus>\<^bsub>V\<^esub> y) = y"
lconf_def: fixes G :: "prog"   and s :: "st"   and vs :: "'a \<Rightarrow> val option"   and Ts :: "'a \<Rightarrow> ty option" shows "G,s\<turnstile>vs[\<Colon>\<preceq>]Ts = (\<forall>n. ! T:Ts n: ? v:vs n: G,s\<turnstile>v\<Colon>\<preceq>T)"
vars_subst_var': fixes lp :: "linear_poly"   and x\<^sub>j :: "nat"   and lp' :: "linear_poly" shows " vars lp - {x\<^sub>j} - vars lp' \<subseteq>s vars (subst_var x\<^sub>j lp' lp) \<subseteq>s vars lp - {x\<^sub>j} \<union> vars lp'"
def_lfp_unfold: fixes h :: "'a"   and f :: "'a \<Rightarrow> 'a" assumes "h \<equiv> lfp f"   and "mono f" shows "h = f h"
simps_1: fixes acc :: "nat"   and n :: "nat" shows "totient_naive 0 acc n = acc"
simps_1: fixes s :: "'a \<Rightarrow> bool option" shows "execute_parallel_plan s [] = s"
unshared_all_unshared: fixes \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set"   and sb :: "'b memref list" assumes "sharing_consistent \<S> \<O> sb" shows "dom \<S> - dom (share sb \<S>) \<subseteq> all_unshared sb"
adjacent_nodes_ss_V: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and u :: "nat" shows "Graph.adjacent_nodes c u \<subseteq> Graph.V c"
wf_rcv_fv_single: fixes V :: "'c set"   and t :: "('d, 'c) Term.term"   and S' :: "('d, 'c) strand_step list" assumes "wf\<^sub>s\<^sub>t V (receive\<langle>t\<rangle>\<^sub>s\<^sub>t # S')" shows "fv t \<subseteq> V"
eq_trans: fixes eq :: "'b \<rightarrow> 'b \<rightarrow> tr"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.Eq_equiv eq"   and "eq\<cdot>x\<cdot>y = TT"   and "eq\<cdot>y\<cdot>z = TT" shows "eq\<cdot>x\<cdot>z = TT"
rep_eq: fixes x :: "('c \<times> 'd) fset" shows "is_fmap x = is_map (fset x)"
enn2real_less: fixes e :: "ennreal"   and r :: "real" assumes "enn2real e < r"   and "e \<noteq> top" shows "e < ennreal r"
monad_stateT': assumes "monad Monad_Overloading.return Monad_Overloading.bind" shows "monad Monad_Overloading.return Monad_Overloading.bind"
stableD: fixes F :: "'c program"   and A :: "'c set" assumes "F \<in> stable A" shows "F \<in> A co A"
2: shows "length is2 = card {i \<in> - A. i < length is1 + length is2}"
WT_converterD_WT: fixes \<I> :: "('a, 'b) \<I>"   and \<I>' :: "('c, 'd) \<I>"   and conv :: "('a, 'b, 'c, 'd) converter"   and q :: "'a" assumes "\<I>, \<I>' \<turnstile>\<^sub>C conv \<surd>"   and "q \<in> outs_\<I> \<I>" shows "\<I>' \<turnstile>g run_converter conv q \<surd>"
lfp_exp_unitary: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>P. unitary P \<Longrightarrow> unitary (t P)" shows "unitary (lfp_exp t)"
redT_upd_\<epsilon>_def: fixes s :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('c \<times> 'a \<Rightarrow>f nat) option) \<times> 'd) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and t :: "'b"   and x' :: "'c"   and m' :: "'d" shows "redT_upd_\<epsilon> s t x' m' = (locks s, (thr s(t \<mapsto> (x', snd (the (thr s t)))), m'), wset s, interrupts s)"
csmall_nstep_Rec2: fixes P :: "'prog"   and \<sigma> :: "'state"   and n :: "nat" shows "csmall_nstep P \<sigma> (Suc n) = {(\<sigma>2, coll). \<exists>\<sigma>1 coll1 coll2. (\<sigma>1, coll1) \<in> csmall P \<sigma> \<and> (\<sigma>2, coll2) \<in> csmall_nstep P \<sigma>1 n \<and> combine coll1 coll2 = coll}"
unRoot_Root: fixes i :: "nat" assumes "i < n" shows "unRoot (Root i) = i"
zero_order_5: fixes n :: "'c multiset" shows "(\<not> {#} \<subset># n) = (n = {#})"
prv_lcnj_imp_in: fixes \<phi>s :: "'fmla list"   and \<phi> :: "'fmla" assumes "set \<phi>s \<subseteq> fmla"   and "\<phi> \<in> set \<phi>s" shows "prv (imp (lcnj \<phi>s) \<phi>)"
xor_def: fixes x :: "'a"   and y :: "'a" shows "x XOR y = x AND NOT y OR NOT x AND y"
filterlim_If: fixes f :: "'b \<Rightarrow> 'c"   and G :: "'c filter"   and F :: "'b filter"   and P :: "'b \<Rightarrow> bool"   and g :: "'b \<Rightarrow> 'c" assumes "filterlim f G (inf F (principal {x. P x}))"   and "filterlim g G (inf F (principal {x. \<not> P x}))" shows "LIM x F. if P x then f x else g x :> G"
integrable_completion: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure" assumes "f \<in> borel_measurable M" shows "integrable (completion M) f = integrable M f"
le_fininf_def: fixes w\<^sub>1 :: "'action list"   and w\<^sub>2 :: "'action stream" shows "w\<^sub>1 \<preceq>\<^sub>F\<^sub>I w\<^sub>2 \<equiv> \<exists>v\<^sub>2. v\<^sub>2 \<le>\<^sub>F\<^sub>I w\<^sub>2 \<and> w\<^sub>1 \<preceq>\<^sub>F v\<^sub>2"
psubset_card_mono: fixes B :: "'g set"   and A :: "'g set" assumes "finite B"   and "A \<subset> B" shows "card A < card B"
diff: shows "A - B \<union> (B - A) \<subseteq> X"
dom_llp1: fixes x :: "'a"   and y :: "'a" assumes "x \<le> d y \<cdot> x" shows "d x \<le> d y"
ennreal_add_eq_top: fixes a :: "ennreal"   and b :: "ennreal" shows "(a + b = \<top>) = (a = \<top> \<or> b = \<top>)"
map2_val_compatI: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and f :: "('a \<times> 'b) list \<Rightarrow> ('a \<times> 'c) list" assumes "comparator comp"   and "\<And>zs. comparator.oalist_inv_raw comp zs \<Longrightarrow> comparator.oalist_inv_raw comp (f zs)"   and "\<And>zs. comparator.oalist_inv_raw comp zs \<Longrightarrow> fst ` set (f zs) \<subseteq> fst ` set zs" shows "comparator.map2_val_compat comp f"
classical_operator_existsI: fixes B :: "'c ell2 \<Rightarrow>\<^sub>C\<^sub>L 'd ell2"   and \<pi> :: "'c \<Rightarrow> 'd option" assumes "\<And>x. B *\<^sub>V ket x = (case \<pi> x of None \<Rightarrow> 0 | Some i \<Rightarrow> ket i)" shows "classical_operator_exists \<pi>"
assms: shows "s \<rightarrow>\<^sub>\<beta> s'"
welldefined_hyp: shows "all_list_constr_R (fst (snd (reduce_system_R p (qs, M_mat_R signs subsets, subsets, signs)))) (length qs)"
simps: fixes "consider" :: "'d \<Rightarrow> bool"   and gpv :: "('e, 'd, 'f) gpv" shows "interaction_bound consider gpv = (\<Squnion>generat\<in>set_spmf (the_gpv gpv). case generat of Pure x \<Rightarrow> 0 | IO out c \<Rightarrow> if consider out then eSuc (\<Squnion>input. interaction_bound consider (c input)) else \<Squnion>input. interaction_bound consider (c input))"
dm_max_elt: fixes r :: "('b \<times> 'b) set"   and x :: "'b"   and M :: "'b multiset" assumes "irrefl r"   and "trans r"   and "x \<in> r \<down>m M" shows "\<exists>y\<in>#M -s r \<down>m M. (x, y) \<in> r"
map_poly_dvd: fixes p :: "real poly"   and q :: "real poly" assumes "map_poly complex_of_real p dvd map_poly complex_of_real q"   and "q \<noteq> 0" shows "p dvd q"
int: shows "integrable (S \<Otimes>\<^sub>M T) f"
lfinite_code_2: fixes x :: "'b"   and xs :: "'b llist" shows "lfinite (x ## xs) = lfinite xs"
step_Normal_elim_cases_2: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and f :: "'c"   and g :: "'b set"   and c :: "('b, 'a, 'c) com"   and s :: "'b"   and u :: "('b, 'a, 'c) com \<times> ('b, 'c) xstate" assumes "\<Gamma>\<turnstile> (Guard f g c, Normal s) \<rightarrow> u" obtains "u = (c, Normal s)" and "s \<in> g"   | "u = (Skip, Fault f)" and "s \<notin> g"
eventually_ball_finite_distrib: fixes A :: "'b set"   and P :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and net :: "'c filter" assumes "finite A" shows "(\<forall>\<^sub>F x in net. \<forall>y\<in>A. P x y) = (\<forall>y\<in>A. \<forall>\<^sub>F x in net. P x y)"
zero_mat_def: fixes nr :: "nat"   and nc :: "nat" shows "0\<^sub>m nr nc \<equiv> mat nr nc (\<lambda>ij. 0::'m)"
not_asym_iff: fixes x :: "'a"   and y :: "'a" assumes "x \<in> X"   and "y \<in> X" shows "(\<not> x \<sqsubset> y) = (y \<sqsubseteq> x)"
cinner_not_same_CBasis: fixes scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and scaleC :: "complex \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and cinner :: "'c \<Rightarrow> 'c \<Rightarrow> complex"   and CBasis :: "'c set"   and u :: "'c"   and v :: "'c" assumes "class.ceuclidean_space scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner CBasis"   and "u \<in> CBasis"   and "v \<in> CBasis"   and "u \<noteq> v" shows "cinner u v = 0"
inf2: fixes B :: "'a"   and k :: "'a"   and b :: "'a"   and a :: "'a" assumes "B \<equiv> k \<sqinter> b" shows "a \<sqinter> B \<equiv> k \<sqinter> (a \<sqinter> b)"
dist_minus: fixes x :: "'a"   and y :: "'a" shows "dist (- x) (- y) = dist x y"
proper_dia_def: fixes p :: "('b, 'c) fm" shows "proper_dia p \<equiv> case p of Pro x \<Rightarrow> Map.empty x | Nom x \<Rightarrow> Map.empty x | \<^bold>\<not> x \<Rightarrow> Map.empty x | fm1 \<^bold>\<or> x \<Rightarrow> Map.empty x | \<^bold>\<diamond> p \<Rightarrow> if \<nexists>a. p = Nom a then Some p else None | \<^bold>@ b x \<Rightarrow> Map.empty x"
fv_subst_bv2_upper_bound: fixes t :: "Core.term"   and lev :: "nat"   and u :: "Core.term" shows "fv (subst_bv2 t lev u) \<subseteq> fv t \<union> fv u"
orthogonal_matrix: fixes Q :: "((real, 'a) vec, 'a) vec" shows "orthogonal_matrix Q = (Finite_Cartesian_Product.transpose Q ** Q = mat 1)"
conjugate_exponent_ennrealI_4: fixes p :: "ennreal"   and q :: "ennreal" assumes "1 / p + 1 / q = 1" shows "q = conjugate_exponent p"
obs_singleton: fixes sourcenode :: "'f \<Rightarrow> 'g"   and targetnode :: "'f \<Rightarrow> 'g"   and kind :: "'f \<Rightarrow> 'h edge_kind"   and valid_edge :: "'f \<Rightarrow> bool"   and Entry :: "'g"   and Def :: "'g \<Rightarrow> 'i set"   and Use :: "'g \<Rightarrow> 'i set"   and state_val :: "'h \<Rightarrow> 'i \<Rightarrow> 'j"   and Exit :: "'g"   and n :: "'g"   and S :: "'g set" assumes "WeakControlDependencePDG sourcenode targetnode kind valid_edge Entry Def Use state_val Exit" shows "(\<exists>m. CFG.obs sourcenode targetnode valid_edge n (WeakControlDependencePDG.PDG_BS_w sourcenode targetnode valid_edge Def Use Exit S) = {m}) \<or> CFG.obs sourcenode targetnode valid_edge n (WeakControlDependencePDG.PDG_BS_w sourcenode targetnode valid_edge Def Use Exit S) = {}"
inj_def: fixes f :: "'c \<Rightarrow> 'd" shows "inj f = (\<forall>x y. f x = f y \<longrightarrow> x = y)"
log_inverse: fixes a :: "real"   and x :: "real" assumes "0 < a"   and "a \<noteq> 1"   and "0 < x" shows "log a (inverse x) = - log a x"
set_intros_2: fixes x22 :: "'a"   and x21 :: "'a"   and x23 :: "'b" shows "x22 \<in> set1_arr (MkArr x21 x22 x23)"
vrange_atD: fixes b :: "V"   and r :: "V" assumes "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> r"   and "vsv r" shows "\<exists>a\<in>elts (\<D>\<^sub>\<circ> r). r\<lparr>a\<rparr> = b"
nonzero_gen: fixes bs :: "('a poly \<times> nat) set" assumes "bs \<subseteq> as" shows "(\<Prod>(a, i)\<in>bs. a) \<noteq> 0"
update_simps_16: fixes V :: "(nat, 'b) mapping"   and s :: "('a, 'b) state" shows "\<T> (\<V>_update V s) = \<T> s"
def_hashmap_size_varType_def: fixes uu :: "varType itself" shows "def_hashmap_size uu \<equiv> 10"
assms_5: shows "emeasure P (space M) \<noteq> \<infinity>"
of_nat_CHAR: shows "of_nat (semiring_char TYPE('a)) = (0::'a)"
subst_ext_iff: fixes \<sigma> :: "'s"   and \<tau> :: "'s" shows "(\<sigma> = \<tau>) = (\<forall>A. A \<cdot>a \<sigma> = A \<cdot>a \<tau>)"
non_inf_def: fixes r :: "('b \<times> 'b) set" shows "non_inf r \<equiv> \<forall>a f. \<exists>i. (f i, f (Suc i)) \<notin> r \<or> (f i, a) \<notin> r"
norm_def: fixes x :: "'b" shows "norm x = sqrt (real_of (inner x x))"
YS1_def: shows "YS1 = {#ys \<in># mset (chop d xs). med ys < x#}"
final_def: fixes e :: "expr" shows "final e \<equiv> (\<exists>v. e = Val v) \<or> (\<exists>r. e = Throw r)"
nneg_const: fixes c :: "'b" shows "nneg (\<lambda>x. c) = ((0::'b) \<le> c)"
chamber_retraction2: fixes C :: "'a set" assumes "C \<in> f \<turnstile> folding_g.\<C>" shows "f ` C = C"
m'_def: shows "m' = root (traverse_tree path' (stern_brocot_iterate_aux I))"
identity\<^sub>2_def: shows "(\<^bold>=) \<equiv> \<lambda>F G. \<^bold>\<forall>x. (\<^bold>\<lambda>y. \<lparr>F,x\<^sup>P,y\<^sup>P\<rparr>) \<^bold>= (\<^bold>\<lambda>y. \<lparr>G,x\<^sup>P,y\<^sup>P\<rparr>) \<^bold>& (\<^bold>\<lambda>y. \<lparr>F,y\<^sup>P,x\<^sup>P\<rparr>) \<^bold>= (\<^bold>\<lambda>y. \<lparr>G,y\<^sup>P,x\<^sup>P\<rparr>)"
mult_row_iarray_def: fixes A :: "'b iarray iarray"   and a :: "nat"   and q :: "'b" shows "mult_row_iarray A a q = IArray.of_fun (\<lambda>n. if n = a then mult_iarray (A !! a) q else A !! n) (IArray.length A)"
continuous_on_component: fixes s :: "'a set"   and f :: "'a \<Rightarrow> ('b, 'c) vec"   and i :: "'c" assumes "continuous_on s f" shows "continuous_on s (\<lambda>x. f x $ i)"
subformulas\<^sub>\<nu>_finite: fixes \<phi> :: "'b ltln" shows "finite (subformulas\<^sub>\<nu> \<phi>)"
comp_arr_ide: fixes a :: "'a"   and f :: "'a" assumes "B.ide a"   and "B.seq f a" shows "f \<cdot>\<^sub>B a = f"
sw_simps3: fixes zs :: "'b"   and xs :: "'b"   and x :: "'c"   and z1 :: "'c"   and z2 :: "'c" assumes "zs \<noteq> xs \<or> x \<notin> {z1, z2}" shows "x @xs[z1 \<and> z2]_zs = x"
post_if_regular_and_not_pre: fixes t :: "('a, 'b, 'c) event list"   and i :: "nat" assumes "regular_event (t ! i)"   and "\<not> prerecording_event t i"   and "i < length t" shows "postrecording_event t i"
cases: fixes x :: "nat \<times> nat \<times> nat" obtains xa :: "nat" and y :: "nat" and z :: "nat" where "x = (xa, y, z)"
pnet_init_not_in_net_tree_ips_not_in_net_ips: fixes s :: "'c net_state"   and np :: "nat \<Rightarrow> ('c, 'd seq_action) automaton"   and p :: "net_tree"   and i :: "nat" assumes "s \<in> init (pnet np p)"   and "i \<notin> net_tree_ips p" shows "i \<notin> net_ips s"
minus_add_distrib: fixes a :: "'a"   and b :: "'a" shows "- (a + b) = - a + - b"
entails_combine: fixes P :: "'c \<Rightarrow> real"   and t :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and R :: "'c \<Rightarrow> real"   and Q :: "'c \<Rightarrow> real"   and S :: "'c \<Rightarrow> real" assumes "P \<tturnstile> t R"   and "Q \<tturnstile> t S"   and "sub_conj t"   and "sound R"   and "sound S" shows "P && Q \<tturnstile> t (R && S)"
ide_hcomp: fixes f :: "'a arrow_of_spans_data"   and g :: "'a arrow_of_spans_data" assumes "ide f"   and "ide g"   and "src f = trg g" shows "ide (f \<star> g)"
split_indicator: fixes P :: "'d \<Rightarrow> bool"   and S :: "'e set"   and x :: "'e" shows "P (indicator S x) = ((x \<in> S \<longrightarrow> P (1::'d)) \<and> (x \<notin> S \<longrightarrow> P (0::'d)))"
wrap_unwrap_ID: shows "wrap oo unwrap oo llast_body = llast_body"
the_br_\<alpha>_def: fixes R :: "('c \<times> 'd) set" shows "the_br_\<alpha> R \<equiv> \<lambda>x. SOME y. (x, y) \<in> R"
Sup_ow_axioms: shows "Sup_ow U\<^sub>1 Sup\<^sub>1"
insert_5: shows "insert s S \<subseteq> carrier (union_ring C)"
tendsto_ln_powr_over_powr': fixes b :: "real"   and a :: "real" assumes "0 < b" shows "((\<lambda>x. ln x powr a / x powr b) \<longlongrightarrow> 0) at_top"
perron_frobenius_positive_ev: assumes "\<And>v. v \<in> local.S \<Longrightarrow> A *v (\<chi>i. complex_of_real (v $h i)) \<noteq> 0" shows "\<exists>v. eigen_vector A v (complex_of_real (HMA_Connect.spectral_radius A)) \<and> real_non_neg_vec v"
t_le: shows "\<forall>t'\<le>t. local.prefixes t' j \<down>"
simps_2: fixes uu :: "'b \<Rightarrow> bool" shows "uu \<Turnstile> \<bottom> = False"
C_eq_iD: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" assumes "separated p"   and "noDenyAll1 p"   and "wellformed_policy1_strong p" shows "Cp (list2FWpolicy (insertDenies p)) = Cp (list2FWpolicy p)"
assms_3: shows "(H * d)\<^sup>+ \<le> - H"
preorder_ow_axioms: shows "preorder_ow U\<^sub>1 le\<^sub>1 ls\<^sub>1"
smult_nsg_sg: fixes H :: "'a set"   and N :: "'a set" assumes "G \<guillemotright> H "   and "G \<triangleright> N" shows "G \<guillemotright> N \<diamondop> H "
assms_1: fixes b :: "'a set" assumes "b \<in># remove1_mset bl \<B>" shows "0 < bl |\<inter>| b"
abs_divide: 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 abs :: "'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.field_abs_sgn divide inverse times one plus zero minus uminus abs sgn" shows "abs (divide a b) = divide (abs a) (abs b)"
positive_p: fixes x :: "'message" shows "0 \<le> M x"