hcju/isabelleps huggingface dataset:
big_subsetI: fixes L :: "'b filter \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'c) set"   and l :: "'b filter \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'c) set"   and L' :: "'d filter \<Rightarrow> ('d \<Rightarrow> 'c) \<Rightarrow> ('d \<Rightarrow> 'c) set"   and l' :: "'d filter \<Rightarrow> ('d \<Rightarrow> 'c) \<Rightarrow> ('d \<Rightarrow> 'c) set"   and Lr :: "'b filter \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> ('b \<Rightarrow> real) set"   and lr :: "'b filter \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> ('b \<Rightarrow> real) set"   and R :: "real \<Rightarrow> real \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "landau_pair L l L' l' Lr lr R"   and "f \<in> L F g" shows "L F f \<subseteq> L F g"
l10_10_spec: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and P' :: "'a"   and P :: "'a"   and A :: "'a"   and B :: "'a"   and Q' :: "'a"   and Q :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.ReflectL Bet Cong P' P A B"   and "Tarski_neutral_dimensionless.ReflectL Bet Cong Q' Q A B" shows "Cong P Q P' Q'"
finite_UnI: fixes F :: "'b set"   and G :: "'b set" assumes "finite F"   and "finite G" shows "finite (F \<union> G)"
pairwise_disjnt_countable: fixes \<N> :: "nat set set" assumes "disjoint \<N>" shows "countable \<N>"
hyps_8: shows "(D' @ TVarB X Q # G)\<turnstile>T2<:T11"
cring_simprules_24: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "x \<otimes> (y \<otimes> z) = y \<otimes> (x \<otimes> z)"
emeasure_density: fixes f :: "'b \<Rightarrow> ennreal"   and M :: "'b measure"   and A :: "'b set" assumes "f \<in> borel_measurable M"   and "A \<in> sets M" shows "emeasure (density M f) A = set_nn_integral M A f"
norm_normalize: fixes x :: "'a" assumes "x \<noteq> (0::'a)" shows "norm (Projections.normalize x) = 1"
Mapping_lookup_fold_upd_set_idle: fixes xs :: "('a \<times> 'b) list"   and as :: "'c"   and Z :: "'b \<Rightarrow> 'a \<Rightarrow> 'c set"   and m :: "('c, 'a) mapping" assumes "{(t, X). (t, X) \<in> set xs \<and> as \<in> Z X t} = {}" shows "Mapping.lookup (fold (\<lambda>(t, X) m. upd_set m (\<lambda>_. t) (Z X t)) xs m) as = Mapping.lookup m as"
iterates_above_ge: fixes lub :: "'b set \<Rightarrow> 'b"   and leq :: "('b \<times> 'b) set"   and f :: "'b \<Rightarrow> 'b"   and y :: "'b"   and a :: "'b" assumes "bourbaki_witt_fixpoint lub leq f"   and "y \<in> bourbaki_witt_fixpoint.iterates_above lub leq f a"   and "a \<in> Field leq" shows "(a, y) \<in> leq"
strategyproofness: shows "pmf (sds R30) c + (pmf (sds R30) b + pmf (sds R30) a) \<le> pmf (sds R41) c + (pmf (sds R41) b + pmf (sds R41) a)"
dij_neq_0: fixes a :: "nat list"   and b :: "nat list"   and i :: "nat"   and j :: "nat" assumes "hlde a b"   and "i < length a"   and "j < length b" shows "Linear_Diophantine_Equations.dij a b i j \<noteq> 0"
supp_eqvt: fixes p :: "perm"   and x :: "'e" shows "p \<bullet> supp x = supp (p \<bullet> x)"
intros: fixes t :: "trm" assumes "\<And>t'. t \<mapsto> t' \<Longrightarrow> SN t'" shows "SN t"
simps: fixes x :: "'b" shows "repeat\<cdot>x = x : repeat\<cdot>x"
iso_is_arr: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "category C"   and "category.iso C f" shows "partial_magma.arr C f"
simps_15: fixes v :: "bexp"   and va :: "'b acom"   and vb :: "'b acom"   and vc :: "'b"   and vd :: "char list"   and ve :: "aexp"   and vf :: "'b" shows "(IF v THEN va ELSE vb {vc} \<sqsubseteq> vd ::= ve {vf}) = False"
valid_Act_fresh: fixes \<alpha> :: "'act"   and P :: "'state"   and x :: "('b, 'pred, 'act) formula" assumes "bn \<alpha> \<sharp>* P" shows "P \<Turnstile> Act \<alpha> x = (\<exists>P'. P \<rightarrow> \<langle>\<alpha>,P'\<rangle> \<and> P' \<Turnstile> x)"
opnet_maintains_dom: fixes \<sigma> :: "nat \<Rightarrow> 'c"   and ns :: "'d net_state"   and a :: "'e node_action"   and \<sigma>' :: "nat \<Rightarrow> 'c"   and ns' :: "'d net_state"   and np :: "nat \<Rightarrow> ((nat \<Rightarrow> 'c) \<times> 'd, 'e seq_action) automaton"   and p :: "net_tree" assumes "((\<sigma>, ns), a, \<sigma>', ns') \<in> automaton.trans (opnet np p)" shows "net_ips ns = net_ips ns'"
EQ: fixes x :: "'b"   and y :: "'b" shows "(cmp x y = EQ) = (x = y)"
add_0_left: fixes a :: "'a" shows "(zero_class.zero::'a) + a = a"
map_graph_edge_preserving: fixes f :: "('d \<times> 'e) set"   and G :: "('f, 'd) labeled_graph" shows "edge_preserving f (edges G) (edges (map_graph f G))"
simps_2: fixes X :: "'b set set"   and x :: "'b set" assumes "SimplicialComplex X" shows "SimplicialComplex.min_maxsimpchain X [x] = SimplicialComplex.maxsimp X x"
one_not_primepow: shows "\<not> primepow (1::'a)"
SINK_mono': fixes f :: "'a \<times> 'a \<Rightarrow> ennreal"   and g :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "f \<le> g" shows "SINK g \<subseteq> SINK f"
nonrec: assumes "phiDefNodes_aux g v unvisited n = {}"   and "n \<notin> set unvisited \<or> v \<in> defs g n" shows "thesis"
Theap_simp: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and a :: "Arity" shows "Theap \<Gamma> e\<cdot>a = (if nonrec \<Gamma> then ccTTree (edom (Aheap \<Gamma> e\<cdot>a)) (\<G>\<^bsub>a\<^esub> e) else ttree_restr (edom (Aheap \<Gamma> e\<cdot>a)) anything)"
distinct_2: fixes lam1' :: "lam"   and lam2' :: "lam"   and name :: "name" shows "App lam1' lam2' \<noteq> Var name"
sumlist_carrier: fixes xs :: "'a vec list" assumes "set xs \<subseteq> carrier_vec n" shows "sumlist xs \<in> carrier_vec n"
nn_integral_fst: fixes f :: "'d \<times> 'c \<Rightarrow> ennreal"   and M1 :: "'d measure" assumes "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M T)" shows "\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>T \<partial>M1 = integral\<^sup>N (M1 \<Otimes>\<^sub>M T) f"
progress_prefix_conv: fixes \<pi> :: "(char list \<times> 'a list) prefix"   and \<sigma> :: "(char list \<times> 'a list) trace"   and \<sigma>' :: "(char list \<times> 'a list) trace"   and \<phi> :: "'a MFOTL.formula" assumes "prefix_of \<pi> \<sigma>"   and "prefix_of \<pi> \<sigma>'" shows "Monitor.progress \<sigma> \<phi> (plen \<pi>) = Monitor.progress \<sigma>' \<phi> (plen \<pi>)"
lm04: fixes P :: "'a \<Rightarrow> bool"   and l :: "'a list" shows "concat (map (\<lambda>n. if P n then [n] else []) l) = concat (map (\<lambda>n. if n \<in> set l then if P n then [n] else [] else []) l)"
ratio_test_convergence: fixes f :: "nat \<Rightarrow> real" assumes "\<forall>\<^sub>F n in sequentially. 0 < f n"   and "1 < liminf (\<lambda>n. ereal (f n / f (Suc n)))" shows "summable f"
c'_def: shows "c' = (MAX n\<in>insert n0 {n0..<n0'}. norm (f n) / norm (g n))"
a0i_inv1_iagreeD: fixes h :: "agent list"   and d :: "'b"   and x :: "'b a0n_state" assumes "set h \<subseteq> good"   and "d \<notin> corrupted x"   and "x \<in> a0i_inv1_iagree" shows "signals x (Commit h d) \<le> signals x (Running h d)"
rel_return_pmf: fixes R :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and x1 :: "'b"   and x2 :: "'d" shows "rel_pmf R (return_pmf x1) (return_pmf x2) = R x1 x2"
elims_2: fixes x :: "'c local.term" assumes "Diag x" obtains "x = \<^bold>\<I>"   | f :: "'c" where "x = \<^bold>\<langle>fa__\<^bold>\<rangle>" and "C.arr fa__"   | f :: "'c" and u :: "'c local.term" where "x = \<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<otimes> u" and "C.arr f \<and> Diag u \<and> u \<noteq> \<^bold>\<I>"
trimono_def: fixes B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" shows "trimono B \<equiv> monotone flatf_ge flatf_ge B \<and> mono B"
has_derivative: fixes g :: "'d \<Rightarrow> 'b"   and g' :: "'d \<Rightarrow> 'b"   and F :: "'d filter" assumes "(g has_derivative g') F" shows "((\<lambda>x. g' (g x)) has_derivative (\<lambda>x. g' (g' x))) F"
msg_distinct_95: fixes X :: "msg"   and Y :: "msg"   and X' :: "ltkey" shows "Enc X Y \<noteq> LtK X'"
suf_comp_not_suf: fixes u :: "'b list"   and v :: "'b list" assumes "u \<bowtie>\<^sub>s v"   and "\<not> v \<le>s u" shows "u <s v"
dg_is_arrD_1: fixes a :: "V"   and b :: "V"   and f :: "V" assumes "f : a \<mapsto>\<^bsub>\<BB>\<^esub> b" shows "f \<in>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>"
power_commutes: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and n :: "nat" assumes "class.monoid_mult one times" shows "times (power.power one times a n) a = times a (power.power one times a n)"
sets_eq: shows "sets (completion (density M f)) = sets (density (completion M) f)"
R_map: fixes r :: "'f"   and m :: "'v" assumes "r \<in> UNIV"   and "m \<in> V" shows "T (r \<cdot> m) = r \<star> T m"
span_insertI: fixes p :: "'a"   and B :: "'a set"   and r :: "'a" assumes "p \<in> ideal B" shows "p \<in> ideal (insert r B)"
cases: fixes a :: "'b multiset" assumes "local.well_base a" obtains M :: "'b multiset" where "a = M" and "\<forall>n. count M n < base"
assms_3: shows "\<forall>rsg\<in>ran \<Gamma> \<union> {[Rule m a]}. \<forall>r\<in>set rsg. (\<nexists>chain. get_action r = Goto chain) \<and> get_action r \<noteq> Unknown"
case_cong: fixes lname :: "lname"   and lname' :: "lname"   and f1 :: "ename \<Rightarrow> 'b"   and g1 :: "ename \<Rightarrow> 'b"   and f2 :: "'b"   and g2 :: "'b" assumes "lname = lname'"   and "\<And>x1. lname' = EName x1 \<Longrightarrow> f1 x1 = g1 x1"   and "lname' = This \<Longrightarrow> f2 = g2" shows "(case lname of EName x1 \<Rightarrow> f1 x1 | This \<Rightarrow> f2) = (case lname' of EName x1 \<Rightarrow> g1 x1 | This \<Rightarrow> g2)"
ss: fixes G :: "'a set" assumes "G \<in> \<G>" shows "G \<subseteq> \<V>"
closure_ofE2: fixes a :: "'c"   and S :: "('c, 'd) eq_object_scheme"   and A :: "'c set" assumes "a \<in> closure_of\<^bsub>S\<^esub> A" obtains a' :: "'c" where "a \<in> carrier S" and "a' \<in> A" and "a .=\<^bsub>S\<^esub> a'"
cod_eqI': fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and f :: "'b" assumes "category C"   and "a \<in> partial_magma.codomains C f" shows "a = partial_magma.cod C f"
geod_mult: fixes n :: "nat"   and a :: "nat"   and k :: "nat" shows "geod n a = geod (n + k) (a * 2 ^ k)"
amult_0_r: fixes z :: "int" shows "ant z * 0 = 0"
fst_netgmap_subnet: fixes sr :: "'e \<Rightarrow> 'c \<times> 'd"   and s2 :: "'e net_state"   and s1 :: "'e net_state" shows "fst (case netgmap sr s1 of (\<sigma>\<^sub>1, ss) \<Rightarrow> case netgmap sr s2 of (\<sigma>\<^sub>2, tt) \<Rightarrow> (\<sigma>\<^sub>1 ++ \<sigma>\<^sub>2, SubnetS ss tt)) = fst (netgmap sr s1) ++ fst (netgmap sr s2)"
language_state_for_inputs_in_language_state: fixes M :: "('d, 'e, 'f) FSM"   and q :: "'f"   and T :: "'d list set" shows "LS\<^sub>i\<^sub>n M q T \<subseteq> LS M q"
awalk_def: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" shows "S.awalk u p v \<equiv> u \<in> pverts sG \<and> set p \<subseteq> parcs sG \<and> pcas u p v"
strict_trans2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "ordering less_eq less"   and "less a b"   and "less_eq b c" shows "less a c"
mul_eq_and_ds_imp_ds: fixes r :: "('a \<times> 'a) set"   and M :: "'a multiset"   and N :: "'a multiset"   and S :: "'a set" assumes "trans r"   and "(M, N) \<in> mul_eq r"   and "set_mset N \<subseteq> r \<down>s S" shows "set_mset M \<subseteq> r \<down>s S"
assms: shows "(\<lambda>x. ereal (u x)) \<longlonglongrightarrow> \<infinity>"
root_cancel1: fixes m :: "nat"   and i :: "nat"   and j :: "nat" shows "FFT.root (2 * m) ^ (i * (2 * j)) = FFT.root m ^ (i * j)"
a_b_step: shows "(a, b) \<in> set (step p \<tau>)"
STCalWB_hasBarbST: fixes Source :: "'a processCalculus"   and Target :: "'b processCalculus"   and SWB :: "('a, 'c) calculusWithBarbs"   and TWB :: "('b, 'c) calculusWithBarbs"   and P :: "('a, 'b) Proc"   and a :: "'c" assumes "encoding_wrt_barbs Source Target SWB TWB" shows "P\<down><STCalWB SWB TWB>a = ((\<exists>S. P = SourceTerm S \<and> S\<down><SWB>a) \<or> (\<exists>T. P = TargetTerm T \<and> T\<down><TWB>a))"
Lp_measurable: fixes f :: "'a \<Rightarrow> real"   and p :: "ennreal"   and M :: "'a measure" assumes "f \<in> lspace M p" shows "f \<in> borel_measurable M"
degree_eq_zeroE: fixes p :: "'c poly" assumes "degree p = 0" obtains a :: "'c" where "p = [:a:]"
AC_assoc: fixes a :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and c :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" shows "a \<Squnion> b \<Squnion> c = (a \<Squnion> b) \<Squnion> c"
la_en'_def: fixes fs :: "local_state \<times> global_state"   and a :: "action" shows "la_en' fs a \<equiv> case la_en fs a of None \<Rightarrow> False | Some b \<Rightarrow> b"
lookup_pair_filter: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and xs :: "('a \<times> 'b) list"   and P :: "'a \<times> 'b \<Rightarrow> bool"   and k :: "'a" assumes "comparator comp"   and "comparator.oalist_inv_raw comp xs" shows "comparator.lookup_pair comp (filter P xs) k = (let v = comparator.lookup_pair comp xs k in if P (k, v) then v else (0::'b))"
ln_upper_5_def: fixes x :: "real" shows "ln_upper_5 x \<equiv> (x\<^sup>2 + 19 * x + 10) * (x - 1) / (3 * (3 * x\<^sup>2 + 6 * x + 1))"
abs_def: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Abs :: "'b set \<Rightarrow> 'c"   and Rep :: "'c \<Rightarrow> 'b set"   and x :: "'b" assumes "quot_type R Abs Rep" shows "quot_type.abs R Abs x = Abs (rel_ext (R x))"
simps_2: fixes a :: "'a"   and as :: "'a list" shows "polynomial (a # as) = (\<lambda>x\<in>carrier R. a \<oplus> x \<otimes> polynomial as x)"
frame_def: fixes a :: "'c \<Longrightarrow> 'b"   and P :: "'b hrel" shows "{a}:[P] = (P \<and> $\<^bold>v\<acute> =\<^sub>u bop (\<triangleleft>\<^bsub>&a\<^esub>) $\<^bold>v $\<^bold>v\<acute>)"
cas: shows "\<forall>i<n. CAs ! i = Cs ! i + poss (AAs ! i)"
assms_1: shows "fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F - set X \<subseteq> subst_domain \<delta>"
65933: fixes x :: "'a"   and y :: "'a" shows "- (- x \<squnion> y) = - - x \<sqinter> - y"
set_spmf_def: fixes p :: "'h spmf" shows "set_spmf p = set_pmf p \<bind> set_option"
intro: fixes hcond :: "'a hrel \<Rightarrow> 'a hrel" assumes "utp_theory hcond"   and "utp_theory_lattice_axioms hcond" shows "utp_theory_lattice hcond"
simps_2: fixes X :: "freemsg"   and Y :: "freemsg" shows "freeleft (MPAIR X Y) = X"
fv_IfThenElse: fixes scrut :: "exp"   and e1 :: "exp"   and e2 :: "exp" shows "fv (scrut ? e1 : e2) = fv scrut \<union> fv e1 \<union> fv e2"
guarded_def: fixes sys :: "'d \<Rightarrow> ('e, 'f) process_term" shows "guarded sys \<equiv> \<forall>X Y. sys X \<noteq> VAR Y"
simps_3: fixes b\<^sub>1 :: "bexp"   and b\<^sub>2 :: "bexp"   and s :: "char list \<Rightarrow> int" shows "bval (And b\<^sub>1 b\<^sub>2) s = (bval b\<^sub>1 s \<and> bval b\<^sub>2 s)"
constant_function_comp_is_constant_seq: fixes a :: "nat \<Rightarrow> int"   and s :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "a \<in> carrier Zp"   and "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>" shows "is_constant_seq Zp (\<cc>\<^bsub>Zp\<^esub> a \<circ> s)"
Rep_map_type: fixes f :: "'b \<Rightarrow> ('c, unit) Old_Datatype.node set"   and A :: "('c, unit) Old_Datatype.node set set"   and xs :: "'b SList.list" assumes "\<And>x. f x \<in> A" shows "Rep_map f xs \<in> list A"
simps_1: fixes f :: "'a \<times> 'c \<Rightarrow> 'a \<times> 'b" shows "map_pair f [] = []"
simps: fixes as :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list" shows "inv_check_left_moving_in_middle (as, lm) (s, l, r) ires = (\<exists>lm1 lm2 r' rn. lm = lm1 @ lm2 \<and> Oc # l = <rev lm1> @ Bk # Bk # ires \<and> r = Oc # Bk # r' \<and> r' = <lm2> @ Bk \<up> rn)"
removel_correct_3: fixes s :: "'s" assumes "invar s"   and "\<alpha> s \<noteq> []" shows "invar (snd (removel s))"
transfer_forall_transfer_2: fixes A :: "'f \<Rightarrow> 'g \<Rightarrow> bool" assumes "right_total A" shows "((A ===> (=)) ===> (\<longrightarrow>)) transfer_forall transfer_forall"
a_subgroup_in_rcosets: shows "(\<Inter>i\<le>n. I (Suc i)) \<in> a_rcosets (\<Inter>i\<le>n. I (Suc i))"
finite_extensionalD_funcset: fixes A :: "'a set"   and B :: "'b set"   and d :: "'b" assumes "finite A"   and "finite B" shows "finite (extensionalD d A \<inter> (A \<rightarrow> B))"
l_set_disconnected_nodes_get_disconnected_nodes_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_set_disconnected_nodes_get_disconnected_nodes_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M ShadowRootClass.known_ptr ShadowRootClass.type_wf get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M Shadow_DOM.parent_child_rel Shadow_DOM.get_child_nodes"
ce_set_lm_3: fixes A :: "nat set" assumes "A \<in> ce_sets" shows "\<exists>p\<in>PrimRec2. A = fn_to_set p"
Maclaurin_all_le_objl: fixes diff :: "nat \<Rightarrow> real \<Rightarrow> real"   and f :: "real \<Rightarrow> real"   and x :: "real"   and n :: "nat" shows "diff 0 = f \<and> (\<forall>m x. (diff m has_real_derivative diff (Suc m) x) (at x)) \<longrightarrow> (\<exists>t. \<bar>t\<bar> \<le> \<bar>x\<bar> \<and> f x = (\<Sum>m<n. diff m 0 / fact m * x ^ m) + diff n t / fact n * x ^ n)"
standardize_apart: fixes C :: "fterm literal set"   and Cs :: "fterm literal set set"   and C' :: "fterm literal set" assumes "C \<in> Cs"   and "var_renaming_of C C'" shows "resolution_step Cs (Cs \<union> {C'})"
cases: fixes a :: "('fenv, 'henv, ('fun, 'dyn) frame) state" assumes "final a" obtains F :: "'fenv" and f :: "'fun" and fd :: "('dyn, 'var, 'fun, 'op, 'opinl) instr fundef" and pc :: "nat" and H :: "'henv" and \<Sigma> :: "'dyn list" where "a = State F H [Frame f pc \<Sigma>]" and "F_get F f = Some fd" and "pc = length (body fd)"
r_step_Id_def: shows "r_step_Id \<equiv> Cn 1 r_prod_encode [r_e2tail, Cn 1 S [Cn 1 r_nth [r_e2xs, Cn 1 r_pdec22 [r_e2i]]]]"
ord_spmf_bind_spmfI1: fixes p :: "'b spmf"   and R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'd spmf"   and q :: "'d spmf" assumes "\<And>x. x \<in> set_spmf p \<Longrightarrow> ord_spmf R (f x) q" shows "ord_spmf R (p \<bind> f) q"
spr_simObsC_def: fixes envObsC :: "'c \<Rightarrow> 'e" shows "spr_simObsC envObsC \<equiv> envObsC \<circ> es \<circ> snd \<circ> ODList.hd"
arrE: fixes f :: "'a" assumes "hom.arr f" shows "\<guillemotleft>f : src e\<^sub>0 \<rightarrow> src e\<^sub>1\<guillemotright>"
listrel_Cons_iff: fixes x :: "'b"   and xs :: "'b list"   and y :: "'c"   and ys :: "'c list"   and r :: "('b \<times> 'c) set" shows "((x # xs, y # ys) \<in> listrel r) = ((x, y) \<in> r \<and> (xs, ys) \<in> listrel r)"
redT_upd_simps: 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 ta :: "('a, 'b, 'c, 'd, 'e, 'f) thread_action"   and x' :: "'c"   and m' :: "'d"   and 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" shows "redT_upd s t ta x' m' s' = (\<exists>ws'. s' = (redT_updLs (locks s) t \<lbrace>ta\<rbrace>\<^bsub>l\<^esub>, (redT_updTs (thr s) \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>(t \<mapsto> (x', redT_updLns (locks s) t (snd (the (thr s t))) \<lbrace>ta\<rbrace>\<^bsub>l\<^esub>)), m'), ws', redT_updIs (interrupts s) \<lbrace>ta\<rbrace>\<^bsub>i\<^esub>) \<and> redT_updWs t (wset s) \<lbrace>ta\<rbrace>\<^bsub>w\<^esub> ws')"
Abs_filter_inverse: fixes y :: "('c \<Rightarrow> bool) \<Rightarrow> bool" assumes "y \<in> {F. is_filter F}" shows "Filter.filter.Rep_filter (Filter.filter.Abs_filter y) = y"
test_absorb2: fixes p :: "'a"   and q :: "'a" assumes "test p"   and "test q" shows "p \<cdot> (p + q) = p"
Rep_name_inject: fixes x :: "name"   and y :: "name" shows "(Rep_name x = Rep_name y) = (x = y)"
ran_m_fmdrop_If: fixes C :: "'b"   and N :: "('b, 'a) fmap" shows "ran_m (fmdrop C N) = (if C \<in># dom_m N then remove1_mset (the (fmlookup N C)) (ran_m N) else ran_m N)"
pratt_complete': fixes p :: "nat" assumes "prime p" shows "\<exists>c. Prime p \<in> set c \<and> valid_cert c \<and> real (length c) \<le> 6 * log 2 (real p) - 4 \<and> (\<forall>x\<in>set c. Pratt_Certificate.size_pratt x \<le> 3 * log 2 (real p))"
eqv: fixes p :: "perm" assumes "p \<bullet> T = S"   and "supp p \<subseteq> atom ` xs \<inter> supp T \<union> atom ` ys \<inter> supp S"   and "p \<bullet> (atom ` xs \<inter> supp T) = atom ` ys \<inter> supp S" shows "p \<bullet> f xs T = f ys S"
CA: shows "\<forall>\<Gamma> \<Delta> A. (( \<Gamma> \<Rightarrow>* \<Delta> \<oplus> A) \<in> derivable' R* \<longrightarrow> ( \<Gamma> \<oplus> A \<Rightarrow>* \<Delta>) \<in> derivable' R*) \<longrightarrow> ( \<Gamma> \<Rightarrow>* \<Delta>) \<in> derivable' R*"
wa_refine_axioms: shows "wa_refine WC WB \<alpha>1"
image_mset_def: fixes f :: "'k \<Rightarrow> 'l" shows "image_mset f = fold_mset (add_mset \<circ> f) {#}"
rep_eq: fixes x :: "'a key_order" shows "key_compare (- x) = (\<lambda>xa y. key_compare x y xa)"
step: fixes y :: "V" assumes "y \<in> elts ya__" shows "rank (x * y) = rank x * rank y"
INF_INT_eq: fixes r :: "'e \<Rightarrow> 'd set"   and S :: "'e set" shows "(INF i\<in>S. (\<lambda>x. x \<in> r i)) = (\<lambda>x. x \<in> \<Inter> (r ` S))"
eval_cases_16: 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 E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" assumes "P,E \<turnstile> \<langle>throw e,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" obtains r :: "nat \<times> char list list" where "e' = Throw r" and "P,E \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>ref r,s'\<rangle>"   | "e' = Throw (addr_of_sys_xcpt NullPointer, [NullPointer])" and "P,E \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>null,s'\<rangle>"   | e'a :: "expr" where "e' = throw e'a" and "P,E \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>throw e'a,s'\<rangle>"
buf: shows "wf_mbufn' \<sigma> P V j n R r buf"
Fvars_PP: fixes t :: "'trm" assumes "t \<in> trm" shows "Fvars (PP t) = FvarsT t"
comp_arr_dom: fixes f :: "'c local.term set"   and a :: "'c local.term set" assumes "\<F>\<^sub>SC.arr f"   and "\<F>\<^sub>SC.dom f = a" shows "\<F>\<^sub>SC.comp f a = f"
relative_toE: fixes P :: "'d set \<Rightarrow> bool"   and U :: "'d set"   and S :: "'d set"   and Q :: "'d set \<Rightarrow> bool" assumes "(P relative_to U) S"   and "\<And>S. P S \<Longrightarrow> Q (U \<inter> S)" shows "Q S"
coeff_linear_power: fixes a :: "'a"   and n :: "nat" shows "coeff ([:a, 1::'a:] ^ n) n = (1::'a)"
integrable_on_subinterval: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and a :: "'a"   and b :: "'a" assumes "f integrable_on S"   and "{a..b} \<subseteq> S" shows "f integrable_on {a..b}"
orientable_12_implies_11: fixes x :: "'a" assumes "x \<sqinter> id x = bot \<and> x \<squnion> id x = - top" shows "(x \<squnion> top) \<sqinter> id (x \<squnion> top) = top \<and> x \<squnion> top \<squnion> id (x \<squnion> top) = top"
less_diff_eq_ennreal: fixes b :: "ennreal"   and c :: "ennreal"   and a :: "ennreal" assumes "b < \<top> \<or> c < \<top>" shows "(a < b - c) = (a + c < b)"