hcju/isabelleps huggingface dataset:
of_nat_floor: fixes r :: "'d" assumes "(0::'d) \<le> r" shows "of_nat (nat \<lfloor>r\<rfloor>) \<le> r"
supp_rel_def: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e" shows "supp_rel R x = {a. infinite {b. \<not> R ((a \<rightleftharpoons> b) \<bullet> x) x}}"
cpu_reg_mod_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and w :: "word32"   and cr :: "CPU_register"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> t1 = cpu_reg_mod w cr s1 \<and> t2 = cpu_reg_mod w cr s2" shows "low_equal t1 t2"
partition_of_unity: fixes A :: "'a set"   and a :: "'a"   and b :: "'a"   and u :: "'a"   and v :: "'a" assumes "ideal R A"   and "a \<in> A"   and "b \<in> carrier R"   and "a \<plusminus> b = 1\<^sub>r"   and "u \<in> carrier R"   and "v \<in> carrier R" shows "pj R A (a \<cdot>\<^sub>r v \<plusminus> b \<cdot>\<^sub>r u) = pj R A u"
normalize_polynomial: fixes K :: "'a set"   and p :: "'a list" assumes "polynomial K p" shows "local.normalize p = p"
map: fixes f :: "'c \<Rightarrow> 'b"   and x1 :: "'c match_expr"   and x2 :: "action" shows "map_rule f (Rule x1 x2) = Rule (map_match_expr f x1) x2"
sort_oalist_def: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and xs :: "('a \<times> 'b) list" assumes "comparator comp" shows "comparator.sort_oalist comp xs = foldr (comparator.update_by_pair comp) xs []"
pair_digraph_axioms: shows "pair_digraph (list_digraph G)"
REC'_def: shows "REC' \<equiv> snd (snd (fix_syn (\<lambda>(COPY', SEND', REC'). (left`?`x \<rightarrow> right`!`x \<rightarrow> COPY', left`?`x \<rightarrow> mid`!`x \<rightarrow> ack \<rightarrow> SEND', mid`?`x \<rightarrow> right`!`x \<rightarrow> ack \<rightarrow> REC'))))"
finite_g_OUT: fixes x :: "'v" shows "d_OUT (\<lambda>(x, y). g (\<langle>y\<rangle>, \<langle>x\<rangle>)) x \<noteq> \<top>"
set_insert_wrt: fixes R :: "('b \<times> 'b) set"   and x :: "'b"   and xs :: "'b list" shows "set (insert_wrt R x xs) = insert x (set xs)"
object_ptr_kinds_M_eq3_h2: shows "object_ptr_kinds h2 = object_ptr_kinds h3"
prems_1: shows "dom (F(x \<mapsto> y)) = dom F2"
l_cancel_inv: fixes h :: "'a"   and a :: "nat" assumes "h \<in> carrier \<G>" shows "\<^bold>g [^] a \<otimes> inv (\<^bold>g [^] a) \<otimes> h = h"
lfinite_induct: fixes xs :: "'a llist"   and P :: "'a llist \<Rightarrow> bool" assumes "lfinite xs"   and "\<And>xs. lnull xs \<Longrightarrow> P xs"   and "\<And>xs. \<lbrakk>lfinite xs; \<not> lnull xs; P (ltl xs)\<rbrakk> \<Longrightarrow> P xs" shows "P xs"
Lim_transform_eventually: fixes f :: "'b \<Rightarrow> 'c"   and l :: "'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "(f \<longlongrightarrow> l) F"   and "\<forall>\<^sub>F x in F. f x = g x" shows "(g \<longlongrightarrow> l) F"
ring_zero: fixes R :: "('h, 'i) Ring_scheme" assumes "Ring R" shows "\<zero>\<^bsub>R\<^esub> \<in> carrier R"
Gcd_nat_abs_eq: fixes K :: "int set" shows "(GCD k\<in>K. nat \<bar>k\<bar>) = nat (Gcd K)"
i_Streams_Connected_Init_def: fixes outS :: "nat \<Rightarrow> 'd message_af"   and inS :: "nat \<Rightarrow> 'd message_af" shows "i_Streams_Connected_Init outS inS \<equiv> inS = outS"
dom_proj_snapshot: fixes PROB :: "(('d, 'e) fmap \<times> ('d, 'e) fmap) set"   and s :: "('d, 'e) fmap" shows "prob_dom (prob_proj PROB (prob_dom (snapshot PROB s))) = prob_dom (snapshot PROB s)"
bind_eq_None_conv: fixes a :: "'c option"   and f :: "'c \<Rightarrow> 'b option" shows "(a \<bind> f = None) = (a = None \<or> f (the a) = None)"
agsprod_assoc: fixes m :: "int"   and n :: "int"   and x :: "ant" assumes "m \<noteq> 0"   and "n \<noteq> 0" shows "m *\<^sub>a (n *\<^sub>a x) = (m * n) *\<^sub>a x"
run_stake: fixes r :: "'action stream"   and p :: "'state"   and k :: "nat" assumes "run r p" shows "path (stake k r) p"
simps: fixes n :: "Integer"   and xs :: "['b]" shows "take\<cdot>n\<cdot>xs = If le\<cdot>n\<cdot>0 then [] else (case xs of [] \<Rightarrow> [] | y : ys \<Rightarrow> y : take\<cdot>(n - 1)\<cdot>ys)"
trd3_simp: fixes a :: "'b"   and b :: "'c"   and c :: "'a" shows "trd3 (a, b, c) = c"
in_arcs_def: fixes G :: "('d, 'c) pre_digraph"   and v :: "'d" shows "in_arcs G v \<equiv> {e \<in> arcs G. head G e = v}"
sum_binomial_times_bernoulli': fixes n :: "nat" shows "(\<Sum>k<n. real (n choose k) * bernoulli k) = (if n = 1 then 1 else 0)"
fun_upd_apply: fixes f :: "'h \<Rightarrow> 'g"   and x :: "'h"   and y :: "'g"   and z :: "'h" shows "(f(x := y)) z = (if z = x then y else f z)"
assms_1: shows "Domainp A = (\<lambda>x. x \<in> U\<^sub>V\<^sub>S)"
select_convs_4: fixes red :: "('f \<times> nat) rgraph"   and black :: "('f, 'd, 'e) lts"   and subs :: "(('f \<times> nat) \<times> 'f \<times> nat) set"   and init_conf :: "('d, 'e) conf"   and confs :: "'f \<times> nat \<Rightarrow> ('d, 'e) conf"   and marked :: "'f \<times> nat \<Rightarrow> bool"   and strengthenings :: "'f \<times> nat \<Rightarrow> ('d \<Rightarrow> 'e) \<Rightarrow> bool"   and more :: "'g" shows "init_conf \<lparr>red = red, black = black, subs = subs, init_conf = init_conf, confs = confs, marked = marked, strengthenings = strengthenings, \<dots> = more\<rparr> = init_conf"
xt1_10: fixes y :: "'b"   and x :: "'b"   and z :: "'b" assumes "y < x"   and "z < y" shows "z < x"
disc_3: fixes x31 :: "'a local.term"   and x32 :: "'a local.term" shows "\<not> is_Prim\<^sub>0 (x31 \<^bold>\<star> x32)"
hyps_2: shows "H - {C} \<turnstile> C IMP A IMP B"
None_eq_map_option_iff: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b option" shows "(None = map_option f x) = (x = None)"
A: fixes i :: "nat" assumes "i < k" shows "0 < bs ! i"
A: shows "P @ P' x @ P'' \<Turnstile> cfs\<box>"
INF_id_eq: fixes A :: "('b \<times> 'c) set" shows "Inf_prod (id ` A) = Inf_prod A"
dom_cod: fixes f :: "'a" assumes "Right.arr f" shows "Right.dom (Right.cod f) = Right.cod f"
has_products_preserved_by_bijection: fixes I :: "'b set"   and \<phi> :: "'b \<Rightarrow> 'c"   and I' :: "'c set" assumes "has_products I"   and "bij_betw \<phi> I I'"   and "I' \<noteq> UNIV" shows "has_products I'"
unity\<^sub>F\<^sub>M\<^sub>C_def: shows "\<I> = \<F>C.mkarr \<^bold>\<I>"
order_gt_0: fixes \<G> :: "'d cyclic_group" assumes "ot_base \<G>" shows "0 < order \<G>"
multi_member_skip: fixes x :: "'a"   and XS :: "'a multiset"   and y :: "'a" assumes "x \<in># XS" shows "x \<in># {#y#} + XS"
insort_key_append1: fixes ys :: "'c list"   and f :: "'c \<Rightarrow> 'a"   and x :: "'c"   and xs :: "'c list" assumes "\<forall>y\<in>set ys. f x < f y" shows "insort_key f x (xs @ ys) = insort_key f x xs @ ys"
convex_hull_list_combination: fixes Vs :: "'a vec list"   and x :: "'a vec"   and y :: "'a vec"   and l :: "'a" assumes "set Vs \<subseteq> carrier_vec n"   and "x \<in> convex_hull_list Vs"   and "y \<in> convex_hull_list Vs"   and "(0::'a) \<le> l"   and "l \<le> (1::'a)" shows "l \<cdot>\<^sub>v x + ((1::'a) - l) \<cdot>\<^sub>v y \<in> convex_hull_list Vs"
feasible_bounded_by: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and P :: "'b \<Rightarrow> real"   and b :: "real" assumes "feasible t"   and "sound P"   and "bounded_by b P" shows "bounded_by b (t P)"
isNode1_def: fixes Kl :: "('d + 'e) list set"   and lab :: "('d + 'e) list \<Rightarrow> ('f, 'd, 'e) F1 + 'g"   and kl :: "('d + 'e) list" shows "isNode1 Kl lab kl = (\<exists>x1. lab kl = Inl x1 \<and> F1set2 x1 = Succ1 Kl kl \<and> F1set3 x1 = Succ2 Kl kl)"
in_lex_prod: fixes a :: "'a"   and b :: "'b"   and a' :: "'a"   and b' :: "'b"   and r :: "('a \<times> 'a) set"   and s :: "('b \<times> 'b) set" shows "(((a, b), a', b') \<in> r <*lex*> s) = ((a, a') \<in> r \<or> a = a' \<and> (b, b') \<in> s)"
2: shows "\<chi>.\<Phi>.Ya.Cop_S.Map (J_C.MkIde D) = D"
le_transI: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and u :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>P. sound P \<Longrightarrow> t P \<tturnstile> u P" shows "le_trans t u"
assms_2: shows "f' : a' \<mapsto>\<^bsub>\<BB>\<^esub> b'"
execn_Normal_elim_cases_2: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and n :: "nat"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>c,Stuck\<rangle> =n\<Rightarrow> t" obtains "t = Stuck"
prod_gen_delta: fixes S :: "'a set"   and a :: "'a"   and b :: "'a \<Rightarrow> 'b"   and c :: "'b" assumes "finite S" shows "(\<Prod>k\<in>S. if k = a then b k else c) = (if a \<in> S then b a * c ^ (card S - 1) else c ^ card S)"
right_idem: fixes a :: "'a"   and b :: "'a" shows "gcd (gcd a b) b = gcd a b"
fmeasurable_Int_fmeasurable: fixes S :: "'a set"   and M :: "'a measure"   and T :: "'a set" assumes "S \<in> fmeasurable M"   and "T \<in> sets M" shows "S \<inter> T \<in> fmeasurable M"
less_eq_cblinfun_def: fixes A :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'b"   and B :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "(A \<le> B) = (\<forall>\<psi>. \<psi> \<bullet>\<^sub>C (A *\<^sub>V \<psi>) \<le> \<psi> \<bullet>\<^sub>C (B *\<^sub>V \<psi>))"
strict_mono_to_nat: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Rep :: "'a \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'a" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "order.strict_mono less (mod_type.to_nat Rep)"
isolated_verts_def: shows "H.isolated_verts \<equiv> {v \<in> verts H. out_arcs H v = {}}"
intvs_decr_h: fixes l :: "int"   and h :: "int" shows "{l..<h - 1} = {l..<h} - {h - 1}"
ffc: shows "f = abelian_monoid.sumlist (module_vec TYPE(int) n) (map (\<lambda>i. of_int (c i) \<cdot>\<^sub>v F ! i) [0..<length F])"
conth: shows "continuous_on (U \<times> {0..1}) h"
simps_9: fixes V :: "'c set"   and uy :: "('d, 'c) Term.term"   and uz :: "('d, 'c) Term.term"   and S :: "('d, 'c) stateful_strand_step list" shows "wf'\<^sub>s\<^sub>s\<^sub>t V (\<langle>uy in uz\<rangle> # S) = wf'\<^sub>s\<^sub>s\<^sub>t V S"
lasso_of_prpl_simps_2: fixes prpl :: "'e list \<times> 'e list" assumes "snd prpl \<noteq> []" shows "lasso_cycle (lasso_of_prpl prpl) = snd prpl"
emeasure_Diff: fixes M :: "'b measure"   and B :: "'b set"   and A :: "'b set" assumes "emeasure M B \<noteq> \<infinity>"   and "A \<in> sets M"   and "B \<in> sets M"   and "B \<subseteq> A" shows "emeasure M (A - B) = emeasure M A - emeasure M B"
exhaust_sel: fixes "thm" :: "thm" obtains "thm = Thm (concl thm)"
less_eq_literal_def: fixes L :: "'b literal"   and M :: "'b literal" shows "(L \<le> M) = (atm_of L < atm_of M \<or> atm_of L \<le> atm_of M \<and> is_neg L \<le> is_neg M)"
primality_test_def: fixes P :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and n :: "nat" shows "primality_test P n = (if n < 3 \<or> even n then return_pmf (n = 2) else pmf_of_set {2..<n} \<bind> (\<lambda>a. return_pmf (P n a)))"
mk_alt_consistency_subset: fixes C :: "('a, 'b) form set set" shows "C \<subseteq> mk_alt_consistency C"
mpt_power: fixes n :: "nat" shows "mpt M (T ^^ n)"
mk_or_atoms: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "atoms_ltln (mk_or \<phi> \<psi>) \<subseteq> atoms_ltln \<phi> \<union> atoms_ltln \<psi>"
bell01_index_4: shows "|\<beta>\<^sub>0\<^sub>1\<rangle> $$ (3, 0) = 0"
path_unique: fixes ab :: "'a set"   and a :: "'a"   and b :: "'a"   and ab' :: "'a set" assumes "path ab a b"   and "path ab' a b" shows "ab = ab'"
ComposeC: fixes T :: "(('fun, 'atom, 'sets) prot_fun, 'c) Term.term list"   and f :: "('fun, 'atom, 'sets) prot_fun"   and M :: "(('fun, 'atom, 'sets) prot_fun, 'c) Term.term set" assumes "length T = arity f"   and "public f"   and "\<And>t. t \<in> set T \<Longrightarrow> M \<turnstile>\<^sub>c t" shows "M \<turnstile>\<^sub>c Fun f T"
in_homE: fixes f :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr"   and a :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr"   and b :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>] b\<guillemotright>" obtains "J_C.arr f" and "J_C.dom f = a" and "J_C.cod f = b"
assms_4: shows "z \<notin> path_image \<gamma>2"
q_def: fixes z :: "nat" shows "q z = (if pA (c_fst z) (c_snd z) = 0 then c_fst z else a)"
iMODb_Max: fixes r :: "nat"   and m :: "nat"   and c :: "nat" shows "Max [ r, mod m, c ] = r + m * c"
set_vwalk_arcs_append2: fixes q :: "'a list"   and p :: "'a list" shows "set (vwalk_arcs q) \<subseteq> set (vwalk_arcs (p @ q))"
emeasure_space_1: fixes M :: "'d measure" assumes "prob_space M" shows "emeasure M (space M) = 1"
translate_no_collision_range: shows "delayed_safe_distance.no_collision {0..} = no_collision_react {\<delta>..}"
rep_eq: fixes x :: "nat"   and xa :: "nat"   and xb :: "nat \<Rightarrow>\<^sub>0 'b" shows "lookup (higherPowers x xa xb) = (\<lambda>x. lookup xb (inc_above x xa x))"
orbit_trans: fixes s :: "'b"   and f :: "'b \<Rightarrow> 'b"   and t :: "'b"   and u :: "'b" assumes "s \<in> orbit f t"   and "t \<in> orbit f u" shows "s \<in> orbit f u"
ag_pOp_commute: fixes A :: "('h, 'i) aGroup_scheme"   and x :: "'h"   and y :: "'h" assumes "aGroup A"   and "x \<in> carrier A"   and "y \<in> carrier A" shows "x \<plusminus>\<^bsub>A\<^esub> y = y \<plusminus>\<^bsub>A\<^esub> x"
Bound: fixes x :: "name"   and a :: "name"   and Q' :: "pi" assumes "x \<sharp> C"   and "Q \<longmapsto> a<\<nu>x> \<prec> Q'" shows "\<exists>P'. P \<Longrightarrow>a<\<nu>x> \<prec> P' \<and> (P', Q') \<in> Rel"
rComm2: fixes \<Psi> :: "'b"   and \<Psi>\<^sub>Q :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and xvec :: "name list"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and Q :: "('a, 'b, 'c) psi"   and K :: "'a"   and Q' :: "('a, 'b, 'c) psi"   and A\<^sub>Q :: "name list"   and C :: "'d" assumes "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "\<And>C. Prop C (\<Psi> \<otimes> \<Psi>\<^sub>Q) P (M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P') A\<^sub>P \<Psi>\<^sub>P"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> K\<lparr>N\<rparr> \<prec> Q'"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>Q"   and "\<And>C. Prop C (\<Psi> \<otimes> \<Psi>\<^sub>P) Q (K\<lparr>N\<rparr> \<prec> Q') A\<^sub>Q \<Psi>\<^sub>Q"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q \<turnstile> M \<leftrightarrow> K"   and "distinct xvec"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* \<Psi>\<^sub>Q"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* M"   and "A\<^sub>P \<sharp>* N"   and "A\<^sub>P \<sharp>* P'"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* Q'"   and "A\<^sub>P \<sharp>* A\<^sub>Q"   and "A\<^sub>P \<sharp>* xvec"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* \<Psi>\<^sub>P"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* N"   and "A\<^sub>Q \<sharp>* P'"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* K"   and "A\<^sub>Q \<sharp>* Q'"   and "A\<^sub>Q \<sharp>* xvec"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* \<Psi>\<^sub>P"   and "xvec \<sharp>* \<Psi>\<^sub>Q"   and "xvec \<sharp>* P"   and "xvec \<sharp>* M"   and "xvec \<sharp>* Q"   and "xvec \<sharp>* K"   and "A\<^sub>P \<sharp>* C"   and "A\<^sub>Q \<sharp>* C"   and "xvec \<sharp>* C" shows "Prop C \<Psi> (P \<parallel> Q) (\<tau> \<prec> \<lparr>\<nu>*xvec\<rparr>P' \<parallel> Q') (A\<^sub>P @ A\<^sub>Q) (\<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q)"
substval: fixes I :: "interp" assumes "usubstappf \<sigma> Denotational_Semantics.allvars \<phi> \<noteq> undeff" shows "valid_in I (the (usubstappf \<sigma> Denotational_Semantics.allvars \<phi>)) = valid_in (USubst.adjoint \<sigma> I \<omega>) \<phi>"
d2_P2: fixes x :: "nat"   and y :: "nat"   and b :: "nat"   and e2 :: "nat"   and r :: "nat" assumes "x < q"   and "y < q"   and "b < q"   and "e2 < q"   and "r < q" shows "((y, b, ((e2 + q - x) mod q * b + q - r) mod q, e2), (x * ((y + q - b) mod q) + q - r) mod q, (x * y + q - (x * ((y + q - b) mod q) + q - r) mod q) mod q) = ((y, b, (e2 * b + q - (x * y + q - (x * ((y + q - b) mod q) + q - r) mod q) mod q) mod q, e2), (x * ((y + q - b) mod q) + q - r) mod q, (x * y + q - (x * ((y + q - b) mod q) + q - r) mod q) mod q)"
const_subterm_subst_var_obtain: fixes c :: "'a"   and t :: "('a, 'c) Term.term"   and \<sigma> :: "'c \<Rightarrow> ('a, 'b) Term.term" assumes "Fun c [] \<sqsubseteq> t \<cdot> \<sigma>"   and "Fun c [] \<notin> subterms t" obtains x :: "'c" where "x \<in> fv t" and "Fun c [] \<sqsubseteq> \<sigma> x"
assms_2: assumes "\<Phi>" shows "m' \<le> c"
hyps_3: assumes "P \<noteq> P''" shows "[b\<frown>b]P \<Longrightarrow>\<^sub>\<tau> P''"
thetaParZOL1_ZOretr: shows "thetaParZOL1 \<subseteq> ZOretr (thetaParZOL1 \<union> ZObis)"
MkArr_Map: fixes Obj :: "'a set"   and Hom :: "'a \<Rightarrow> 'a \<Rightarrow> 'b set"   and Id :: "'a \<Rightarrow> 'b"   and Comp :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "('a, 'b) concrete_category.arr" assumes "concrete_category Obj Hom Id Comp"   and "partial_magma.arr (concrete_category.COMP Obj Hom Comp) f" shows "concrete_category.MkArr (concrete_category.Dom f) (concrete_category.Cod f) (concrete_category.Map f) = f"
freshAbs_vsubstAbs: fixes A :: "('d, 'e, 'varSort, 'var, 'f) qAbs set"   and zs :: "'varSort"   and z :: "'var"   and ys :: "'varSort"   and y1 :: "'var"   and y :: "'var" assumes "goodAbs A" shows "freshAbs zs z (A $[y1 // y]_ys) = ((zs = ys \<and> z = y \<or> freshAbs zs z A) \<and> (freshAbs ys y A \<or> zs \<noteq> ys \<or> z \<noteq> y1))"
ccINF_commute: fixes A :: "'d set"   and B :: "'e set"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'a" assumes "countable A"   and "countable B" shows "(INF i\<in>A. Inf (f i ` B)) = (INF j\<in>B. INF i\<in>A. f i j)"
Subspace_dim_lt: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and U :: "'b set" assumes "FinDimVectorSpace smult V"   and "RModule UNIV smult U \<and> U \<subseteq> V"   and "U \<noteq> V" shows "R_scalar_mult.dim_R UNIV smult U < R_scalar_mult.dim_R UNIV smult V"
Q3'_a_def: shows "Q3'_a \<equiv> [: \<lambda>(stk, mrk). {(stk', mrk'). stk \<noteq> [] \<and> (\<exists>y. (hd stk, y) \<in> next \<and> y \<notin> mrk \<and> stk' = y # stk \<and> mrk' = mrk \<union> {y})} :]"
finitely_generated_ideal_def: fixes I :: "'b set" shows "finitely_generated_ideal I = (ring_class.ideal I \<and> (\<exists>S. finite S \<and> ideal_generated S = I))"
cfg\<^sub>1: shows "cfg\<^sub>1 = (c\<^sub>1, s)"
bisimilar_def: fixes a :: "('a, 'b) Stream"   and b :: "('a, 'c) Stream" shows "(a \<approx> b) = (unstream\<cdot>a = unstream\<cdot>b \<and> a \<noteq> \<bottom> \<and> b \<noteq> \<bottom>)"
bot_least: fixes a :: "'a" shows "\<top> \<le> a"
simps_2: fixes P :: "'g \<Rightarrow> bool"   and x :: "'g"   and xs :: "'g list" shows "find P (x # xs) = (if P x then Some x else find P xs)"
tdconc: fixes xs :: "int list" shows "\<exists>ys. xs = take_decr xs @ ys"
fun_le_cons: fixes a :: "val \<times> val"   and t1 :: "(val \<times> val) list"   and t2 :: "(val \<times> val) list" assumes "a # t1 \<lesssim> t2" shows "t1 \<lesssim> t2"
le_sup_iff: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "(x \<sqinter> y \<le> z) = (x \<le> z \<and> y \<le> z)"
ab_ge_b2: shows "b\<^sup>2 \<le> a * b"
inject: fixes x2 :: "'a"   and y2 :: "'a" shows "(Some x2 = Some y2) = (x2 = y2)"
P2R_R2P: fixes r :: "('c \<times> 'd) set" shows "P2R (R2P r) = r"
poly_root_induct_alt: fixes Q :: "'a poly \<Rightarrow> bool"   and p :: "'a poly" assumes "Q 0"   and "\<And>p. (\<And>a. poly p a \<noteq> (0::'a)) \<Longrightarrow> Q p"   and "\<And>a p. Q p \<Longrightarrow> Q ([:- a, 1::'a:] * p)" shows "Q p"
hyps: shows "foldl (\<circ>) id fs = foldr (\<circ>) fs id"
same_length_different: fixes xs :: "'d list"   and ys :: "'d list" assumes "xs \<noteq> ys"   and "length xs = length ys" shows "\<exists>pre x xs' y ys'. x \<noteq> y \<and> xs = pre @ [x] @ xs' \<and> ys = pre @ [y] @ ys'"
design_iso_point_rep_num_eq: fixes \<V> :: "'c set"   and \<B> :: "'c set multiset"   and \<V>' :: "'d set"   and \<B>' :: "'d set multiset"   and bij_map :: "'c \<Rightarrow> 'd"   and p :: "'c" assumes "design_isomorphism \<V> \<B> \<V>' \<B>' bij_map"   and "p \<in> \<V>" shows "\<B> rep p = \<B>' rep bij_map 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 backward_slice :: "'g set \<Rightarrow> 'g set"   and n :: "'g"   and S :: "'g set" assumes "BackwardSlice sourcenode targetnode kind valid_edge Entry Def Use state_val backward_slice" shows "card (CFG.obs sourcenode targetnode valid_edge n (backward_slice S)) \<le> 1"
cases: 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 a1 :: "char list"   and a2 :: "char list list" assumes "Subobjs P a1 a2" obtains C :: "char list" and Cs :: "char list list" where "a1 = Ca__" and "a2 = Csa__" and "Subobjs\<^sub>R P Ca__ Csa__"   | C :: "char list" and C' :: "char list" and D :: "char list" and Cs :: "char list list" where "a1 = C" and "a2 = Cs" and "P \<turnstile> C \<preceq>\<^sup>* C'" and "P \<turnstile> C' \<prec>\<^sub>S D" and "Subobjs\<^sub>R P D Cs"
du: shows "d * u = - A $ from_nat k $ from_nat k"
with_bot_def: fixes A :: "'a set" shows "A\<^sub>\<bottom> = {None} \<union> Some ` A"
comm_ring_1_axioms: shows "class.comm_ring_1 (*) (1::'d) (+) (0::'d) (-) uminus"
AR_from_Un_Int_local: fixes S :: "'a set"   and T :: "'a set" assumes "closedin (top_of_set (S \<union> T)) S"   and "closedin (top_of_set (S \<union> T)) T"   and "AR (S \<union> T)"   and "AR (S \<inter> T)" shows "AR S"
H_seq: fixes r :: "('b \<times> 'b) set"   and y :: "('b \<times> 'b) set"   and q :: "('b \<times> 'b) set"   and p :: "('b \<times> 'b) set"   and x :: "('b \<times> 'b) set" assumes "rel_kat.H r y q"   and "rel_kat.H p x r" shows "rel_kat.H p (x ; y) q"
Parallel_Strong_Soundness_Seq: fixes Ts :: "('b ann_com option \<times> 'b set) list"   and b :: "'b"   and i :: "nat"   and c0 :: "'b ann_com"   and c1 :: "'b ann_com" assumes "\<forall>i<length Ts. if OG_Hoare.com (Ts ! i) = None then b \<in> OG_Hoare.post (Ts ! i) else b \<in> pre (the (OG_Hoare.com (Ts ! i))) \<and> \<turnstile> the (OG_Hoare.com (Ts ! i))  OG_Hoare.post (Ts ! i)"   and "OG_Hoare.com (Ts ! i) = Some (AnnSeq c0 c1)"   and "i < length Ts"   and "interfree Ts" shows "(\<forall>ia<length Ts. if OG_Hoare.com (Ts[i := (Some c0, pre c1)] ! ia) = None then b \<in> OG_Hoare.post (Ts[i := (Some c0, pre c1)] ! ia) else b \<in> pre (the (OG_Hoare.com (Ts[i := (Some c0, pre c1)] ! ia))) \<and> \<turnstile> the (OG_Hoare.com (Ts[i := (Some c0, pre c1)] ! ia))  OG_Hoare.post (Ts[i := (Some c0, pre c1)] ! ia)) \<and> interfree (Ts[i := (Some c0, pre c1)])"
in_nv2T: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and x :: "Preliminaries.var"   and T :: "'d trm" assumes "ProblemIkPol wtFsym wtPsym arOf resOf parOf \<Phi> infTp" shows "(x \<in> ProblemIkPol.nv2T T) = (T = Var x)"
interior_subset: fixes S :: "'c set" shows "interior S \<subseteq> S"