hcju/isabelleps huggingface dataset:
take_bit_signed_take_bit: fixes plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and one :: "'k"   and divide :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and modulo :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and bit :: "'k \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'k \<Rightarrow> 'k"   and drop_bit :: "nat \<Rightarrow> 'k \<Rightarrow> 'k"   and take_bit :: "nat \<Rightarrow> 'k \<Rightarrow> 'k"   and "and" :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and or :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and xor :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and mask :: "nat \<Rightarrow> 'k"   and uminus :: "'k \<Rightarrow> 'k"   and not :: "'k \<Rightarrow> 'k"   and m :: "nat"   and n :: "nat"   and a :: "'k" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not"   and "m \<le> Suc n" shows "take_bit m (ring_bit_operations.signed_take_bit zero times one bit take_bit or mask not n a) = take_bit m a"
pdivmod_via_pseudo_divmod: fixes f :: "'b poly"   and g :: "'b poly" shows "(f div g, f mod g) = (if g = 0 then (0, f) else let ilc = inverse (lead_coeff g); h = smult ilc g; (q, r) = pseudo_divmod f h in (smult ilc q, r))"
rw2: shows "foldl (echelon_form_of_column_k bezout) (A, 0) [0..<Suc k] = (echelon_form_of_upt_k A k bezout, if \<forall>m. is_zero_row_upt_k m (Suc k) (echelon_form_of_upt_k A k bezout) then 0 else mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n (Suc k) (echelon_form_of_upt_k A k bezout)) + 1)"
complete_base: fixes n :: "nat"   and E :: "'a set"   and Us :: "'a list" assumes "dimension n K E"   and "local.independent K Us"   and "set Us \<subseteq> E" shows "\<exists>Vs. length (Vs @ Us) = n \<and> local.independent K (Vs @ Us) \<and> Span K (Vs @ Us) = E"
ccache_approx_trans: fixes A :: "((nat \<times> (nat \<Rightarrow> 'a option)) \<times> 'a \<aproc>) set"   and B :: "((nat \<times> (nat \<Rightarrow> 'a option)) \<times> 'a \<aproc>) set"   and C :: "((nat \<times> (nat \<Rightarrow> 'a option)) \<times> 'a \<aproc>) set" assumes "A \<lessapprox> B"   and "B \<lessapprox> C" shows "A \<lessapprox> C"
Ri_weaken: fixes x :: "'b"   and x' :: "'b"   and y' :: "'c"   and y :: "'c"   and z :: "'c" assumes "x \<le> x'"   and "y' \<le> y" shows "Ri x y z \<le> Ri x' y' z"
4: shows "stake (Suc i) w\<^sub>1 \<preceq>\<^sub>F\<^sub>I w\<^sub>2"
SUP_eq: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<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 A :: "'d set"   and B :: "'e set"   and f :: "'d \<Rightarrow> 'b"   and g :: "'e \<Rightarrow> 'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>i. i \<in> A \<Longrightarrow> \<exists>j\<in>B. less_eq (f i) (g j)"   and "\<And>j. j \<in> B \<Longrightarrow> \<exists>i\<in>A. less_eq (g j) (f i)" shows "Sup (f ` A) = Sup (g ` B)"
inner_add_left: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner (plus x y) z = inner x z + inner y z"
mono_inv: fixes f :: "'a \<Rightarrow> 'b" assumes "mono f"   and "bij f" shows "mono (inv f)"
simps_18: fixes x21 :: "'g"   and x22 :: "'i" shows "generat_outs (IO x21 x22) = {x21}"
hm_it_init_def: fixes ht :: "('a, 'b) hashtable" shows "hm_it_init ht \<equiv> Array.len (the_array ht) \<bind> (\<lambda>n. if n = 0 then return (0, [], ht) else hm_it_adjust (n - 1) ht \<bind> (\<lambda>i. Array.nth (the_array ht) i \<bind> (\<lambda>l. return (i, l, ht))))"
splus_mono: fixes v :: "('a, 'b) pp"   and w :: "('a, 'b) pp"   and t :: "('a, 'b) pp" assumes "le_of_nat_term_order to (v, the_min) (w, the_min)" shows "le_of_nat_term_order to (local.punit.splus t v, the_min) (local.punit.splus t w, the_min)"
substitution_axioms: shows "substitution (\<cdot>a) id_subst (\<odot>) renamings_apart atm_of_atms"
R2_def: fixes M :: "'grp \<times> 'grp"   and v :: "bool" shows "R2 M v = (let (m0, m1) = M in sample_uniform (order \<G>) \<bind> (\<lambda>a. sample_uniform (order \<G>) \<bind> (\<lambda>b. let c\<^sub>v = a * b mod order \<G> in sample_uniform (order \<G>) \<bind> (\<lambda>c\<^sub>v'. sample_uniform_units (order \<G>) \<bind> (\<lambda>r0. sample_uniform_units (order \<G>) \<bind> (\<lambda>s0. let w0 = (\<^bold>g [^] a) [^] s0 \<otimes> \<^bold>g [^] r0; z = (\<^bold>g [^] c\<^sub>v') [^] s0 \<otimes> (\<^bold>g [^] b) [^] r0 in sample_uniform_units (order \<G>) \<bind> (\<lambda>r1. sample_uniform_units (order \<G>) \<bind> (\<lambda>s1. let w1 = (\<^bold>g [^] a) [^] s1 \<otimes> \<^bold>g [^] r1; z' = (\<^bold>g [^] c\<^sub>v) [^] s1 \<otimes> (\<^bold>g [^] b) [^] r1; enc_m = z \<otimes> (if v then m0 else m1); enc_m' = z' \<otimes> (if v then m1 else m0) in return_spmf (v, \<^bold>g [^] a, \<^bold>g [^] b, \<^bold>g [^] c\<^sub>v, w0, enc_m, w1, enc_m')))))))))"
fps_ginv: fixes a :: "'a fps"   and b :: "'a fps" assumes "fps_nth a 0 = (0::'a)"   and "fps_nth a 1 \<noteq> (0::'a)" shows "fps_ginv b a oo a = b"
ms_cons: fixes e1 :: "exp"   and e2 :: "exp"   and e3 :: "exp" assumes "e1 \<longrightarrow> e2"   and "e2 \<longrightarrow>* e3" shows "e1 \<longrightarrow>* e3"
sub_distrib_pconjD: fixes prog :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real"   and Q :: "'a \<Rightarrow> real" assumes "sub_distrib_pconj prog"   and "unitary P"   and "unitary Q" shows "wlp prog P && wp prog Q \<tturnstile> wp prog (P && Q)"
CLS_eqI: fixes f :: "'a"   and g :: "'a" assumes "B.ide f" shows "(\<lbrakk>\<lbrakk>f\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>g\<rbrakk>\<rbrakk>) = (f \<cong>\<^sub>B g)"
P_new_1: shows "v0 \<leadsto>P_new\<leadsto>\<^bsub>G\<^esub> v1"
distinct_entries: fixes t :: "('d, 'e) RBT.rbt" shows "distinct (map fst (RBT.entries t))"
PQ': shows "P' * Q' = 1\<^sub>m (n - 1)"
fstar_def: fixes f :: "'b \<Rightarrow> 'b" shows "fstar f = \<Sqinter> range (fpower f)"
AL_update3: fixes L :: "('b \<times> 'a) list"   and a :: "'b"   and v :: "'a"   and b :: "'b"   and X :: "'a option" assumes "L[a\<mapsto>v]\<down>b = X"   and "a \<noteq> b" shows "L\<down>b = X"
prems_2: fixes b :: "bool" shows "ordertype (X b) VWF \<le> (SUP \<xi>\<in>elts \<gamma>. \<xi>)"
max_prob_viterbi: fixes s :: "'s"   and os :: "'t list" assumes "s \<in> \<S>" shows "(MAX t\<in>\<S>. viterbi_prob s t os) = max_prob s os"
drop_take_drop_unsplit: fixes i :: "nat"   and j :: "nat"   and l :: "'b list" assumes "i \<le> j" shows "drop i (take j l) @ drop j l = drop i l"
unit_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b arrow_of_spans_data" assumes "span_bicategory C prj0 prj1" shows "span_bicategory.unit C prj0 prj1 a \<equiv> \<lparr>Chn = prj0 (Chn a) (Chn a), Dom = Dom (span_bicategory.hcomp C prj0 prj1 a a), Cod = Cod a\<rparr>"
sub_deg0: fixes g :: "nat \<Rightarrow> 'a"   and f :: "nat \<Rightarrow> 'a" assumes "g \<in> carrier P"   and "f \<in> carrier P"   and "g \<noteq> \<zero>\<^bsub>P\<^esub>"   and "f \<noteq> \<zero>\<^bsub>P\<^esub>" shows "deg R (Cring_Poly.compose R f g) = deg R f * deg R g"
is_Ref_def: fixes v :: "'a val" shows "is_Ref v \<equiv> v = Null \<or> is_Addr v"
redT_new_thread_ts_Some: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t' :: "'t"   and x :: "'x"   and m'' :: "'m" assumes "s -t\<triangleright>ta\<rightarrow> s'"   and "NewThread t' x m'' \<in> set \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>"   and "wset_thread_ok (wset s) (thr s)" shows "thr s' t' = \<lfloor>(x, no_wait_locks)\<rfloor>"
intros_3: fixes M :: "'a measure"   and N :: "'a measure" assumes "space M = space N"   and "sets M = sets N"   and "emeasure M \<le> emeasure N" shows "M \<le> N"
Min_gr_iff: fixes A :: "'a word set"   and x :: "'a word" assumes "finite A"   and "A \<noteq> {}" shows "(x <s signed.Min A) = (\<forall>a\<in>A. x <s a)"
simps_1: fixes v :: "event_data list"   and x :: "nat" shows "Formula.eval_trm v (trm.Var x) = v ! x"
lnth_lappend: fixes xs :: "'c llist"   and ys :: "'c llist"   and n :: "nat" shows "lnth (lappend xs ys) n = (if enat n < llength xs then lnth xs n else lnth ys (n - the_enat (llength xs)))"
siso_cont_indis: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and c :: "('d, 'c, 'a) cmd"   and s :: "'b"   and t :: "'b"   and i :: "nat" assumes "PL_Indis cval indis"   and "PL_Indis.siso aval tval cval indis c"   and "(s, t) \<in> indis"   and "i < PL.brn c" shows "(PL.eff aval tval cval c s i, PL.eff aval tval cval c t i) \<in> indis \<and> PL.wt aval tval cval c s i = PL.wt aval tval cval c t i \<and> PL.cont aval tval cval c s i = PL.cont aval tval cval c t i"
alpha_tAct: fixes f1 :: "'d"   and \<alpha>1 :: "'c"   and t1 :: "('a, 'b, 'c, 'd) FL_Formula.Tree"   and f2 :: "'d"   and \<alpha>2 :: "'c"   and t2 :: "('a, 'b, 'c, 'd) FL_Formula.Tree" shows "(FL_Formula.Tree.tAct f1 \<alpha>1 t1 =\<^sub>\<alpha> FL_Formula.Tree.tAct f2 \<alpha>2 t2) = (f1 = f2 \<and> (\<exists>p. (bn \<alpha>1, t1) \<approx>set (=\<^sub>\<alpha>) supp_rel (=\<^sub>\<alpha>) p (bn \<alpha>2, t2) \<and> (bn \<alpha>1, \<alpha>1) \<approx>set (=) supp p (bn \<alpha>2, \<alpha>2)))"
plossless_id_converter: fixes \<I> :: "('i, 'j) \<I>" shows "plossless_converter \<I> \<I> 1\<^sub>C"
inner_commute: fixes x :: "'a"   and y :: "'a" shows "inner_field x y = cnj_class.cnj (inner_field y x)"
AA: shows "swapcols I (Suc I) A \<in> carrier_mat n n"
g'_def: shows "g' \<equiv> g +++ pg +++ cp +++ reversepath pg"
sec_id_identified_4: fixes n :: "nat"   and m :: "nat" shows "PubK {Tok_PriK n, Sec_PriK m, Rev_PriK} = PubK {Sec_PriK m, Tok_PriK n, Rev_PriK}"
simps_6: fixes Fstd_empty :: "'a"   and Fstd_get :: "'a \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e) Std.instr fundef option"   and Fstd_add :: "'a \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e) Std.instr fundef \<Rightarrow> 'a"   and Fstd_to_list :: "'a \<Rightarrow> ('b \<times> ('c, 'd, 'b, 'e) Std.instr fundef) list"   and Finca_empty :: "'f"   and Finca_get :: "'f \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e, 'g) Inca.instr fundef option"   and Finca_add :: "'f \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e, 'g) Inca.instr fundef \<Rightarrow> 'f"   and Finca_to_list :: "'f \<Rightarrow> ('b \<times> ('c, 'd, 'b, 'e, 'g) Inca.instr fundef) list"   and heap_empty :: "'h"   and heap_get :: "'h \<Rightarrow> 'd \<times> 'c \<Rightarrow> 'c option"   and heap_add :: "'h \<Rightarrow> 'd \<times> 'c \<Rightarrow> 'c \<Rightarrow> 'h"   and heap_to_list :: "'h \<Rightarrow> (('d \<times> 'c) \<times> 'c) list"   and is_true :: "'c \<Rightarrow> bool"   and is_false :: "'c \<Rightarrow> bool"   and \<OO>\<pp> :: "'e \<Rightarrow> 'c list \<Rightarrow> 'c"   and \<AA>\<rr>\<ii>\<tt>\<yy> :: "'e \<Rightarrow> nat"   and \<II>\<nn>\<ll>\<OO>\<pp> :: "'g \<Rightarrow> 'c list \<Rightarrow> 'c"   and \<II>\<nn>\<ll> :: "'e \<Rightarrow> 'c list \<Rightarrow> 'g option"   and \<II>\<ss>\<II>\<nn>\<ll> :: "'g \<Rightarrow> 'c list \<Rightarrow> bool"   and \<DD>\<ee>\<II>\<nn>\<ll> :: "'g \<Rightarrow> 'e"   and opinl :: "'g" assumes "std_inca_simulation Fstd_empty Fstd_get Fstd_add Fstd_to_list Finca_empty Finca_get Finca_add Finca_to_list heap_empty heap_get heap_add heap_to_list is_true is_false \<OO>\<pp> \<AA>\<rr>\<ii>\<tt>\<yy> \<II>\<nn>\<ll>\<OO>\<pp> \<II>\<nn>\<ll> \<II>\<ss>\<II>\<nn>\<ll> \<DD>\<ee>\<II>\<nn>\<ll>" shows "std_inca_simulation.norm_instr \<DD>\<ee>\<II>\<nn>\<ll> (IOpInl opinl) = Std.instr.IOp (\<DD>\<ee>\<II>\<nn>\<ll> opinl)"
cases: fixes a1 :: "value"   and a2 :: "value" assumes "a1 \<approx>\<^sub>e a2" obtains ts :: "value list" and us :: "value list" and name :: "name" where "a1 = Vconstr name__ ts" and "a2 = Vconstr name__ us" and "list_all2 erelated ts us"   | cs :: "(Term_Class.term \<times> sterm) list" and \<Gamma>\<^sub>1 :: "(name, value) fmap" and \<Gamma>\<^sub>2 :: "(name, value) fmap" where "a1 = Vabs cs \<Gamma>\<^sub>1__" and "a2 = Vabs cs \<Gamma>\<^sub>2__" and "fmrel_on_fset (ids (Sabs cs)) erelated \<Gamma>\<^sub>1__ \<Gamma>\<^sub>2__"   | \<Gamma>\<^sub>1 :: "(name, value) fmap" and \<Gamma>\<^sub>2 :: "(name, value) fmap" and css :: "(name, (Term_Class.term \<times> sterm) list) fmap" and name :: "name" where "a1 = Vrecabs css name \<Gamma>\<^sub>1" and "a2 = Vrecabs css name \<Gamma>\<^sub>2" and "pred_fmap (\<lambda>cs. fmrel_on_fset (ids (Sabs cs)) erelated \<Gamma>\<^sub>1 \<Gamma>\<^sub>2) css"
stop_at_3: fixes c :: "nat \<Rightarrow>f value option"   and r :: "nat \<Rightarrow>f value option"   and t :: "(String.literal \<times> value list) list" shows "\<not> obtains 1 c drinks2 3 r t"
assms_2: shows "k \<in> fst ` set (list_of_oalist_tc xs) \<union> fst ` set (list_of_oalist_tc ys)"
Cno: fixes y :: "complex" assumes "y \<in> convex hull {a, b, c}" shows "cmod y < C"
trans: shows "(s1, s2) \<midarrow>a\<midarrow>composition\<longrightarrow> (t1, t2)"
strategies_continue: fixes v :: "'a"   and w :: "'a"   and \<sigma> :: "'a \<Rightarrow> 'a" assumes "v \<in> S"   and "v \<rightarrow> w"   and "v \<in> VV p \<Longrightarrow> \<sigma> v = w"   and "\<sigma> \<in> good v" shows "\<sigma> \<in> good w"
push: fixes f :: "nat \<times> nat \<Rightarrow> 'capacity"   and l :: "nat \<Rightarrow> nat"   and e :: "nat \<times> nat" assumes "Height_Bounded_Labeling c s t f l"   and "push_precond f l e" shows "((push_effect f e, l), f, l) \<in> gap_algo_rel"
S_1: fixes a :: "'a" assumes "a \<in> A" shows "a \<in> S a"
h': shows "h \<turnstile> (case parent_opt of None \<Rightarrow> return () | Some parent \<Rightarrow> remove_child parent node_ptr) \<rightarrow>\<^sub>h h'"
make'_def: fixes i :: "integer"   and f :: "'b \<Rightarrow> 'a" shows "Array.make' i f = Array.make (nat_of_integer i) (f \<circ> of_nat)"
intro: fixes get_ancestors :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_ancestors_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_root_node_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "get_ancestors = l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_ancestors get_parent"   and "get_ancestors_locs = l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_ancestors_locs get_parent_locs"   and "get_root_node = l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_root_node get_parent"   and "get_root_node_locs = l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_root_node_locs get_parent_locs" shows "l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_parent get_parent_locs get_ancestors get_ancestors_locs get_root_node get_root_node_locs"
assms: shows "\<forall>x\<in>set xs. \<forall>y\<in>set ys. x \<le> y"
elims: fixes x :: "thread"   and y :: "nat" assumes "ni_weight x = y" obtains "x = INIT" and "y = 0"   | v :: "nat" where "x = WAIT v__" and "y = 1"   | v :: "nat" where "x = HOLD va__" and "y = 1"   | v :: "nat" where "x = REL v" and "y = 1"
integral_lborel: fixes f :: "'b \<Rightarrow> 'c" assumes "integrable lborel f" shows "integral UNIV f = integral\<^sup>L lborel f"
leq: shows "P,C \<turnstile> Cs \<sqsubset>\<^sup>1 Ds"
val_nscal_ge_selfTr: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a"   and n :: "nat" assumes "valuation K v"   and "x \<in> carrier K"   and "0 \<le> v x" shows "v x \<le> v ( n \<times>\<^bsub>K\<^esub> x)"
hyps_11: shows "A\<^sub>P \<sharp>* K"
rel_nebset_def: shows "rel_nebset \<equiv> map_fun id (map_fun set_nebset (map_fun set_nebset id)) rel_set"
has_contour_integral: fixes f :: "complex \<Rightarrow> complex"   and i :: "complex"   and g :: "real \<Rightarrow> complex" shows "(f has_contour_integral i) g = ((\<lambda>x. f (g x) * vector_derivative g (at x)) has_integral i) {0..1}"
sumAssocLeft: fixes Rel :: "(ccs \<times> ccs) set"   and P :: "ccs"   and Q :: "ccs"   and R :: "ccs" assumes "Id \<subseteq> Rel" shows "P \<oplus> Q \<oplus> R \<leadsto>[Rel] P \<oplus> (Q \<oplus> R)"
L6: shows "A0 B C0 CongA A' B' C'"
assms: shows "s' \<in> g_orbital f G U S t\<^sub>0 s"
stationary_distribution_unique: fixes A :: "'a st_mat" assumes "fixed_mat.irreducible (st_mat A)" shows "\<exists>!v. A *st v = v"
concat_appr: fixes xsi :: "'b list list"   and xs :: "real list set list" assumes "(xsi, xs) \<in> \<langle>appr_rell\<rangle>list_rel" shows "(concat_appr ops xsi, concat ` listset xs) \<in> appr_rell"
inject: fixes x :: "'f"   and ya :: "'f" shows "(WriterT x = WriterT ya) = (x = ya)"
holomorphic_on_exp': fixes f :: "complex \<Rightarrow> complex"   and s :: "complex set" assumes "f holomorphic_on s" shows "(\<lambda>x. exp (f x)) holomorphic_on s"
intros: fixes a :: "'a"   and \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and \<epsilon> :: "ennreal" assumes "a \<in> A \<Gamma>"   and "a \<notin> \<E>\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f)"   and "d_OUT f a < weight \<Gamma> a"   and "\<epsilon> < weight \<Gamma> a - d_OUT f a" shows "hindrance_by \<Gamma> f \<epsilon>"
times_divide_times_eq: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and x :: "'d"   and y :: "'d"   and z :: "'d"   and w :: "'d" assumes "class.field divide inverse times one plus zero minus uminus" shows "times (divide x y) (divide z w) = divide (times x z) (times y w)"
max_ipv4_addr_def: shows "max_ipv4_addr \<equiv> ipv4addr_of_nat (2 ^ 32 - 1)"
disjoint_add_mset_2: fixes A :: "'a multiset"   and b :: "'a"   and B :: "'a multiset" shows "({#} = A \<inter># add_mset b B) = (b \<notin># A \<and> {#} = A \<inter># B)"
cong_1: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and b :: "'a"   and c :: "'a" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "unique_euclidean_semiring.cong modulo b c one"
path_component_refl: fixes x :: "'c"   and S :: "'c set" assumes "x \<in> S" shows "path_component S x x"
pderiv_power: fixes p :: "'a poly"   and n :: "nat" shows "pderiv (p ^ n) = smult (of_nat n) (p ^ (n - 1)) * pderiv p"
Liminf_le_Limsup: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> 'b" assumes "F \<noteq> bot" shows "Liminf F f \<le> Limsup F f"
simps_2: fixes c1 :: "('g, 'h, 'i, 'j) com"   and c2 :: "('g, 'h, 'i, 'j) com" shows "decompose_com (c1 ;; c2) = {(c, \<lambda>t. ictxt t ;; c2, \<lambda>t. fctxt t @ [c2]) |c fctxt ictxt. (c, ictxt, fctxt) \<in> decompose_com c1}"
alpha_trans: fixes X :: "('c, 'd, 'varSort, 'var, 'e) qTerm"   and X' :: "('c, 'd, 'varSort, 'var, 'e) qTerm"   and X'' :: "('c, 'd, 'varSort, 'var, 'e) qTerm" assumes "qGood X \<or> qGood X' \<or> qGood X''"   and "X #= X'"   and "X' #= X''" shows "X #= X''"
SubstVar_axioms: shows "SubstVar TYPE('a) subst_var"
typ_of1_eta_red_step: fixes t :: "Core.term"   and \<tau> :: "typ"   and \<tau>' :: "typ" assumes "\<not> is_dependent t"   and "typ_of (Abs \<tau> (t $ Bv 0)) = Some (\<tau> \<rightarrow> \<tau>')" shows "typ_of (decr 0 t) = Some (\<tau> \<rightarrow> \<tau>')"
Ax_def: shows "Ax \<equiv> {(xa, x). Axp xa x}"
rep_eq: fixes x :: "'b" shows "fpxs_nth (fpxs_const x) = (\<lambda>n. if n = 0 then x else (0::'b))"
g_value_image_filter_impl: shows "map_value_image_filter m1.\<alpha> m1.invar m2.\<alpha> m2.invar g_value_image_filter"
object_ptr_kinds_eq: shows "object_ptr_kinds h' = object_ptr_kinds h |\<union>| {|cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_document_ptr|}"
countable_reachable: fixes s :: "'s" shows "countable (acc `` {s})"
lambda_system_sets: fixes x :: "'b set"   and \<Omega> :: "'b set"   and M :: "'b set set"   and f :: "'b set \<Rightarrow> ennreal" assumes "x \<in> lambda_system \<Omega> M f" shows "x \<in> M"
vrestr_of_set: fixes g :: "'a \<Rightarrow> 'a"   and A :: "'a set" assumes "g \<in> M \<rightarrow>\<^sub>M M"   and "A \<in> sets M" shows "g --` A = g -` A \<inter> space M"
Inheritance: assumes "G \<turnstile>Method old member_of C" shows "thesis"
Mp_rel_i_Mp_to_int_poly_i: fixes f :: "'i list"   and F :: "int poly" assumes "Mp_rel_i f F" shows "Mp (to_int_poly_i ff_ops f) = to_int_poly_i ff_ops f"
zero_le_power2: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq zero (power.power one times a 2)"
plus_pratt2_def: fixes x :: "pratt2"   and y :: "pratt2" shows "x + y \<equiv> pratt2_max x y"
nn_integral_count_space': fixes A :: "'c set"   and B :: "'c set"   and f :: "'c \<Rightarrow> ennreal" assumes "finite A"   and "\<And>x. \<lbrakk>x \<in> B; x \<notin> A\<rbrakk> \<Longrightarrow> f x = 0"   and "A \<subseteq> B" shows "integral\<^sup>N (count_space B) f = sum f A"
norm_bound_bridge: fixes A :: "'a mat"   and b :: "real" assumes "\<forall>e\<in>elements_mat A. norm e \<le> b" shows "local.norm_bound A b"
no_moves1: fixes tl1 :: "'tl1"   and s1' :: "'s1" shows "\<not> s1 -1-tl1\<rightarrow> s1'"
WHILET_rule: fixes R :: "('b \<times> 'b) set"   and I :: "'b \<Rightarrow> bool"   and s :: "'b"   and b :: "'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b nres"   and \<Phi> :: "'b \<Rightarrow> bool" assumes "wf R"   and "I s"   and "\<And>s. \<lbrakk>I s; b s\<rbrakk> \<Longrightarrow> f s \<le> SPEC (\<lambda>s'. I s' \<and> (s', s) \<in> R)"   and "\<And>s. \<lbrakk>I s; \<not> b s\<rbrakk> \<Longrightarrow> \<Phi> s" shows "WHILE\<^sub>T b f s \<le> SPEC \<Phi>"