hcju/isabelleps huggingface dataset:
mono_sinvar: fixes N :: "'a set"   and E :: "('a \<times> 'a) set"   and E' :: "('a \<times> 'a) set"   and nP :: "'a \<Rightarrow> 'b" assumes "wf_graph \<lparr>nodes = N, edges = E\<rparr>"   and "E' \<subseteq> E"   and "sinvar \<lparr>nodes = N, edges = E\<rparr> nP" shows "sinvar \<lparr>nodes = N, edges = E'\<rparr> nP"
add_sup_distrib_right: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "sup a b + c = sup (a + c) (b + c)"
vector_all_zero_from_index3: fixes i :: "'c"   and A :: "(('d, 'e) vec, 'c) vec"   and k :: "'e" shows "(\<exists>m>i. A $ m $ k \<noteq> (0::'d)) = (\<not> vector_all_zero_from_index (mod_type_class.to_nat i + 1, vec_to_iarray (column k A)))"
SUP_subset_mono: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and A :: "'c set"   and B :: "'c set"   and f :: "'c \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "A \<subseteq> B"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq (f x) (g x)" shows "less_eq (Sup (f ` A)) (Sup (g ` B))"
INT: shows "(\<lambda>n. LINT y|count_space C. D y n) \<longlonglongrightarrow> 0"
\<psi>_def: fixes x :: "'c"   and h :: "'d" shows "F.\<psi> x h = transformation_by_components.map (\<cdot>\<^sub>C) (\<cdot>\<^sub>C) (F \<circ> F.G) F.\<epsilon>o x \<cdot>\<^sub>C F h"
InstanceOf_\<tau>red1t_xt: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e :: "'addr expr1"   and xs :: "'addr val list"   and e' :: "'addr expr1"   and xs' :: "'addr val list"   and T :: "ty" assumes "\<tau>red1gt uf P t h (e, xs) (e', xs')" shows "\<tau>red1gt uf P t h (e instanceof T, xs) (e' instanceof T, xs')"
Stable_eq_stable: fixes F :: "'c program"   and A :: "'c set" shows "(F \<in> Stable A) = (F \<in> stable (reachable F \<inter> A))"
map_of_eq_Some_iff: fixes xys :: "('a \<times> 'b) list"   and x :: "'a"   and y :: "'b" assumes "distinct (map fst xys)" shows "(map_of xys x = Some y) = ((x, y) \<in> set xys)"
min_arg_not_ge_2: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and m :: "'c"   and n :: "'c" assumes "class.linorder less_eq less" shows "(\<not> less (ord.min less_eq m n) n) = (ord.min less_eq m n = n)"
select_convs_4: fixes ISignature :: "('c, 'a) Signature"   and ICategory :: "('d, 'b) Category"   and ITypes :: "'c \<Rightarrow> 'd"   and IFunctions :: "'a \<Rightarrow> 'b"   and more :: "'e" shows "IFunctions \<lparr>ISignature = ISignature, ICategory = ICategory, ITypes = ITypes, IFunctions = IFunctions, \<dots> = more\<rparr> = IFunctions"
iprefix_def: fixes xs :: "'b list"   and f :: "nat \<Rightarrow> 'b" shows "xs \<sqsubseteq> f \<equiv> \<exists>g. f = xs \<frown> g"
hoare3o_valid_GC: fixes P :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" assumes "\<Turnstile>\<^sub>3\<^sub>' {P} c { Q}" shows "\<Turnstile>\<^sub>3\<^sub>' {P} c { Q \<and>* (\<lambda>s. True)}"
map_\<I>_id: fixes \<I> :: "('a, 'b) \<I>" shows "map_\<I> id id \<I> = \<I>"
P_def: shows "P = (\<lambda>m. if prime (m + 1) \<and> m dvd 2 * n then 1 / (real m + 1) else 0)"
tree_take_smaller: fixes i :: "nat"   and t :: "'r tree"   and u :: "'r tree"   and j :: "nat" assumes "tree_take i\<cdot>t = tree_take i\<cdot>u"   and "j \<le> i" shows "tree_take j\<cdot>t = tree_take j\<cdot>u"
env_restr_belowI: fixes S :: "'c set"   and m\<^sub>1 :: "'c \<Rightarrow> 'd"   and m\<^sub>2 :: "'c \<Rightarrow> 'd" assumes "\<And>x. x \<in> S \<Longrightarrow> m\<^sub>1 x \<sqsubseteq> m\<^sub>2 x" shows "m\<^sub>1 f|` S \<sqsubseteq> m\<^sub>2 f|` S"
algebraic_consistent: fixes K :: "(('a list \<times> nat) multiset \<Rightarrow> 'a) set" assumes "subring K L" shows "Id.S.algebraic = (\<lambda>Ka x. \<not> ring.transcendental (L\<lparr>carrier := K\<rparr>) Ka x)"
l_get_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M ShadowRootClass.type_wf ShadowRootClass.known_ptr DocumentClass.type_wf DocumentClass.known_ptr Shadow_DOM.get_child_nodes Shadow_DOM.get_child_nodes_locs Core_DOM_Functions.get_child_nodes Core_DOM_Functions.get_child_nodes_locs"
C_def: shows "C = (\<lambda>t. ball t (ee t / 3)) ` {0..1}"
suffix2: shows "suffix k2 w \<Turnstile>\<^sub>n af (formula n) (w [index n \<rightarrow> k2])[X]\<^sub>\<nu>"
unbal_ins_Unbal: fixes bal_i :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and n :: "nat"   and d :: "nat"   and x :: "'b"   and t :: "'b tree"   and t' :: "'b tree" assumes "RBTi1 TYPE('b) bal_i"   and "RBTi1.ins bal_i n d x t = Unbal t'" shows "\<not> bal_i (n + 1) (height t' + d)"
sup_assertion: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top"   and "x \<in> mbt_algebra.assertion one times inf less_eq dual top"   and "y \<in> mbt_algebra.assertion one times inf less_eq dual top" shows "sup x y \<in> mbt_algebra.assertion one times inf less_eq dual top"
correctCompositionDiffLevelsA12: shows "correctCompositionDiffLevels sA12"
pass\<^sub>1_struct: fixes hs1 :: "'b tree"   and x :: "'b"   and hs :: "'b tree" shows "\<exists>l a r. pass\<^sub>1 \<langle>hs1, x, hs\<rangle> = \<langle>l, a, r\<rangle>"
comp_reduce: fixes f :: "'a"   and g :: "'a"   and k :: "'a"   and h :: "'a" assumes "f \<cdot> g = k"   and "seq f g"   and "seq g h" shows "f \<cdot> g \<cdot> h = k \<cdot> h"
null_sets_completionI: fixes N :: "'b set"   and M :: "'b measure" assumes "N \<in> null_sets M" shows "N \<in> null_sets (completion M)"
lcs_adds_fun: fixes s :: "'a \<Rightarrow> 'b"   and u :: "'a \<Rightarrow> 'b"   and t :: "'a \<Rightarrow> 'b" assumes "s adds u"   and "t adds u" shows "lcs s t adds u"
lpair_def: fixes lp :: "'d \<times> ('fun, 'e) Term.term \<times> ('fun, 'e) Term.term" shows "lpair lp \<equiv> case lp of (i, p) \<Rightarrow> (i, pair p)"
adds_term_def: fixes u :: "'t"   and v :: "'t" shows "(u adds\<^sub>t v) = (component_of_term u = component_of_term v \<and> pp_of_term u adds pp_of_term v)"
inject_1: fixes x1 :: "fresh_t"   and y1 :: "fresh_t" shows "(nonce_fresh x1 = nonce_fresh y1) = (x1 = y1)"
sAsA'': fixes n :: "nat" assumes "n < length qs" shows "s_A n = s_A' n"
s_r': shows "s r' = Some (\<sigma>', \<tau>', VE v)"
st2vars1': shows "st2.vars1' = {0..<Suc k}"
scale_minus_right: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'b"   and x :: "'c" assumes "Modules.module scale" shows "scale a (- x) = - scale a x"
\<tau>_1: shows "\<tau> \<in> ptrm 0"
cases: fixes a1 :: "char list exp list"   and a2 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and a3 :: "char list exp list"   and a4 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list" assumes "P \<turnstile> \<langle>a1,a2\<rangle> [\<rightarrow>] \<langle>a3,a4\<rangle>" obtains e :: "char list exp" and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" and e' :: "char list exp" and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" and es :: "char list exp list" where "a1 = e # es__" and "a2 = s__" and "a3 = e' # es__" and "a4 = s'__" and "P \<turnstile> \<langle>e,s__\<rangle> \<rightarrow> \<langle>e',s'__\<rangle>"   | es :: "char list exp list" and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" and es' :: "char list exp list" and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" and v :: "val" where "a1 = Val v # es" and "a2 = s" and "a3 = Val v # es'" and "a4 = s'" and "P \<turnstile> \<langle>es,s\<rangle> [\<rightarrow>] \<langle>es',s'\<rangle>"
ffun_lens_def: fixes i :: "'b" shows "ffun_lens i = \<lparr>lens_get = \<lambda>s. s(i)\<^sub>f, lens_put = \<lambda>s. ffun_upd s i\<rparr>"
ceuclidean_all_zero_iff: fixes x :: "'a" shows "(\<forall>u\<in>CBasis. \<langle>x, u\<rangle> = 0) = (x = (0::'a))"
simps_1: fixes x :: "name" shows "frees (Free x) = {|x|}"
nodes_target: fixes p :: "'state"   and r :: "'transition list" assumes "p \<in> nodes"   and "path r p" shows "target r p \<in> nodes"
DF_simps_2: fixes l :: "nat" shows "DF (Suc l) = F l (DF l)"
Sup_eq_Inf: 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 :: "'b set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup A = Inf {b. \<forall>a\<in>A. less_eq a b}"
trans_T: fixes P :: "'a process"   and Q :: "'a process"   and S :: "'a process" assumes "P \<sqsubseteq>\<^sub>T Q"   and "Q \<sqsubseteq>\<^sub>T S" shows "P \<sqsubseteq>\<^sub>T S"
add_diff_eq: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.group_add minus plus zero uminus" shows "plus a (minus b c) = minus (plus a b) c"
chine_hcomp_arr_src: fixes \<mu> :: "'a arrow_of_spans_data" assumes "arr \<mu>" shows "chine_hcomp \<mu> (src \<mu>) = \<langle>Chn \<mu> \<cdot> \<p>\<^sub>1[Leg0 (Dom \<mu>), C.cod (Leg0 (Dom \<mu>))] \<lbrakk>Leg0 (Cod \<mu>), C.cod (Leg0 (Dom \<mu>))\<rbrakk> \<p>\<^sub>0[Leg0 (Dom \<mu>), C.cod (Leg0 (Dom \<mu>))]\<rangle>"
chop_min: fixes i :: "nat_int"   and j :: "nat_int"   and k :: "nat_int" shows "N_Chop(i,j,k) \<and> consec j k \<longrightarrow> minimum i = minimum j"
orderE: fixes a :: "'a"   and b :: "'a" assumes "a \<preceq> b" obtains "a = ordered_powerprod_lin.min a b"
markedElementsAppend: fixes M1 :: "('b \<times> bool) list"   and M2 :: "('b \<times> bool) list" shows "markedElements (M1 @ M2) = markedElements M1 @ markedElements M2"
dsum_pOp_mem: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme"   and X :: "'c \<Rightarrow> 'd"   and Y :: "'c \<Rightarrow> 'd" assumes "\<forall>k\<in>I. aGroup (A k)"   and "X \<in> carr_dsumag I A"   and "Y \<in> carr_dsumag I A" shows "prod_pOp I A X Y \<in> carr_dsumag I A"
worec_fixpoint1: fixes H :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c"   and i :: "'a" assumes "local.adm_wo H" shows "local.worec H i = H (local.worec H) i"
erlang_distributed_iff: fixes l :: "real"   and X :: "'a \<Rightarrow> real"   and k :: "nat" assumes "0 < l" shows "distributed M lborel X (\<lambda>x. ennreal (erlang_density k l x)) = (random_variable borel X \<and> 0 < l \<and> (\<forall>a\<ge>0. prob {x \<in> space M. X x \<le> a} = erlang_CDF k l a))"
E_def: shows "E \<equiv> {(u, v). c (u, v) \<noteq> (0::'capacity)}"
dsafe_Const: fixes r :: "real" shows "dsafe (Const r)"
CSP2_ok_b: fixes P :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool"   and a :: "('c, 'd) alpha_rp_scheme"   and b :: "('c, 'd) alpha_rp_scheme" assumes "P (a, b\<lparr>ok := True\<rparr>) \<or> P (a, b\<lparr>ok := False\<rparr>)" shows "CSP2 P (a, b\<lparr>ok := True\<rparr>)"
isPartition_wrtI: fixes lo :: "nat"   and p :: "nat"   and R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list"   and hi :: "nat" assumes "\<And>i. \<lbrakk>lo \<le> i; i < p\<rbrakk> \<Longrightarrow> R (xs ! i) (xs ! p)"   and "\<And>j. \<lbrakk>p < j; j \<le> hi\<rbrakk> \<Longrightarrow> R (xs ! p) (xs ! j)" shows "isPartition_wrt R xs lo hi p"
get_ancestors_si_ptrs_in_heap: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_ancestors_si :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_ancestors_si_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_root_node_si :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_root_node_si_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and get_disconnected_document :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_disconnected_document_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and h :: "(_) heap"   and ptr :: "(_) object_ptr"   and ancestors :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr" assumes "l_get_root_node_si_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr known_ptrs get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_host get_host_locs get_ancestors_si get_ancestors_si_locs get_root_node_si get_root_node_si_locs get_disconnected_nodes get_disconnected_nodes_locs get_shadow_root get_shadow_root_locs get_tag_name heap_is_wellformed parent_child_rel heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_document get_disconnected_document_locs"   and "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_ancestors_si ptr \<rightarrow>\<^sub>r ancestors"   and "ptr' \<in> set ancestors" shows "ptr' |\<in>| object_ptr_kinds h"
countable_reachable: fixes K :: "'d \<Rightarrow> 'd pmf"   and s :: "'d" shows "countable ((SIGMA x:UNIV. set_pmf (K x))\<^sup>* `` {s})"
DC_idem: fixes a :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" shows "a \<Sqinter> a = a"
Send_L2: fixes return :: "nat \<Rightarrow> 'a Frame list"   and send :: "nat \<Rightarrow> 'a Frame list"   and get :: "nat \<Rightarrow> nat list"   and activation :: "nat \<Rightarrow> nat list"   and t :: "nat" assumes "Send return send get activation"   and "activation t \<noteq> []"   and "return t \<noteq> []" shows "send t \<noteq> []"
epsclosure_steps: fixes A :: "'b \<times> ('c option \<Rightarrow> 'b \<Rightarrow> 'b set) \<times> ('b \<Rightarrow> bool)"   and w :: "'c list" shows "NAe.steps A w O (eps A)\<^sup>* = NAe.steps A w"
f: fixes i :: "nat" shows "AE x in M. f (Suc i) x \<le> f i x"
iffs: fixes Leg0 :: "'b"   and Leg1 :: "'b"   and more :: "'c"   and Leg0' :: "'b"   and Leg1' :: "'b"   and more' :: "'c" shows "(\<lparr>Leg0 = Leg0, Leg1 = Leg1, \<dots> = more\<rparr> = \<lparr>Leg0 = Leg0', Leg1 = Leg1', \<dots> = more'\<rparr>) = (Leg0 = Leg0' \<and> Leg1 = Leg1' \<and> more = more')"
size_greater_than_median_of_medians: fixes xs :: "'a list" shows "size (filter_mset ((<) (median (map median (chop 5 xs)))) (mset xs)) \<le> nat \<lceil>7 / 10 * real (length xs) + 3\<rceil>"
starlike_def: fixes S :: "'b set" shows "starlike S = (\<exists>a\<in>S. \<forall>x\<in>S. closed_segment a x \<subseteq> S)"
length_dominant_term_ms: fixes F :: "'a ms" shows "length (snd (dominant_term F)) = Suc (expansion_level TYPE('a))"
cases: fixes a1 :: "('a, 'b) sequent list \<times> ('a, 'b) sequent"   and a2 :: "('a, 'b) form"   and a3 :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set" assumes "rightPrincipal a1 a2 a3" obtains C :: "('a, 'b) sequent" and A :: "('a, 'b) form" and Ps :: "('a, 'b) sequent list" and R :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set" where "a1 = (Ps, C)" and "a2 = A" and "a3 = R" and "C = ( \<Empt> \<Rightarrow>* \<LM> A  \<RM>)" and "(Ps, C) \<in> R"
no_black_refsD: fixes s :: "'e process_name \<Rightarrow> ('f, 'e, 'g, 'h) Model.local_state"   and r :: "'h" assumes "no_black_refs s" shows "\<not> black r s"
pr_index_enumerator_increase2: fixes n :: "nat"   and m :: "nat" shows "pr_index_enumerator n m < pr_index_enumerator n (m + 1)"
select_convs_1: fixes init :: "'b set"   and trans :: "('b \<times> 'b) set"   and obs :: "'b \<Rightarrow> 'c"   and more :: "'d" shows "obs \<lparr>init = init, trans = trans, obs = obs, \<dots> = more\<rparr> = obs"
snd_sym_preproc_aux_complete: fixes fs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and vs :: "'t list"   and gs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and u :: "'t"   and ks :: "'t list"   and g :: "'t \<Rightarrow>\<^sub>0 'c" assumes "\<And>u' g'. \<lbrakk>u' \<in> Keys (set fs); u' \<notin> set vs; g' \<in> set gs; lt g' adds\<^sub>t u'\<rbrakk> \<Longrightarrow> monom_mult (1::'c) (pp_of_term u' - lp g') g' \<in> set fs"   and "u \<in> Keys (set (snd (sym_preproc_aux gs ks (vs, fs))))"   and "g \<in> set gs"   and "lt g adds\<^sub>t u" shows "monom_mult (1::'c) (pp_of_term u - lp g) g \<in> set (snd (sym_preproc_aux gs ks (vs, fs)))"
fun_def: shows "fun = (\<lambda>U. if \<exists>s. crit U s then insert (SOME s. crit U s) U else U)"
forest_add_edge: fixes G :: "('a, 'b) graph"   and a :: "'a"   and b :: "'a"   and w :: "'b" assumes "forest G"   and "a \<in> nodes G"   and "b \<in> nodes G"   and "\<not> nodes_connected G a b" shows "forest (add_edge a w b G)"
lm084: fixes N :: "'a set"   and G :: "'b list"   and bids :: "'a \<times> 'b set \<Rightarrow> 'c" assumes "N \<noteq> {}"   and "finite N"   and "distinct G"   and "set G \<noteq> {}" shows "winningAllocationRel N (set G) (\<inter>) bids (set (allAllocationsAlg N G)) \<noteq> {}"
monic_degree_map_poly_hom: fixes p :: "'q mod_ring poly" assumes "monic p" shows "degree (map_poly rebase p) = degree p"
set_take_if_index: fixes xs :: "'c list"   and x :: "'c"   and i :: "nat" assumes "index xs x < i"   and "i \<le> length xs" shows "x \<in> set (take i xs)"
thetaIfW_def: shows "thetaIfW \<equiv> {(if tst then c1 else c2, if tst then d1 else d2) |tst c1 c2 d1 d2. compatTst tst \<and> c1 \<approx>w d1 \<and> c2 \<approx>w d2}"
spec_ops_def: fixes ops :: "('a \<times> 'b) list"   and deps :: "'b \<Rightarrow> 'a set" shows "spec_ops ops deps \<equiv> sorted (map fst ops) \<and> distinct (map fst ops) \<and> (\<forall>oid oper ref. (oid, oper) \<in> set ops \<and> ref \<in> deps oper \<longrightarrow> ref < oid)"
that: fixes T :: "'a set" assumes "open T"   and "x \<in> T"   and "T \<subseteq> S" shows "thesis"
legendre_aux_def: fixes x :: "real"   and p :: "nat" shows "legendre_aux x p = (if prime p then \<Sum>m | 0 < m \<and> real (p ^ m) \<le> x. nat \<lfloor>x / real (p ^ m)\<rfloor> else 0)"
Rep_upset_inverse: fixes x :: "'a upset" shows "Abs_upset (Rep_upset x) = x"
ext_inject: fixes prg :: "prog"   and cls :: "qtname"   and lcl :: "lenv"   and more :: "'b"   and prg' :: "prog"   and cls' :: "qtname"   and lcl' :: "lenv"   and more' :: "'b" shows "(\<lparr>prg = prg, cls = cls, lcl = lcl, \<dots> = more\<rparr> = \<lparr>prg = prg', cls = cls', lcl = lcl', \<dots> = more'\<rparr>) = (prg = prg' \<and> cls = cls' \<and> lcl = lcl' \<and> more = more')"
zdvd_mult_cancel1: fixes m :: "int"   and n :: "int" assumes "m \<noteq> 0" shows "(m * n dvd m) = (\<bar>n\<bar> = 1)"
map\<^sub>0_simps_3: fixes a :: "'a" assumes "C.obj a" shows "trg\<^sub>D (map\<^sub>0 a) = map\<^sub>0 a"
snd_Gauss_Jordan_iarrays_PA_eq: fixes A :: "'b iarray iarray" shows "snd (Gauss_Jordan_iarrays_PA A) = Gauss_Jordan_iarrays A"
spick_def: fixes w :: "bool stream"   and n :: "nat"   and v :: "bool" shows "spick w n v = stake n w @- v ## sconst True"
canonical_proj_ring_hom: fixes I :: "'a set"   and J :: "'a set" assumes "ideal I R"   and "ideal J R" shows "ring_hom_ring R (RDirProd (R Quot I) (R Quot J)) (canonical_proj I J)"
simps_4: fixes n :: "nat"   and k :: "nat"   and T :: "POPLmark.type"   and U :: "POPLmark.type" shows "\<up>\<^sub>\<tau> n k (\<forall><:T. U) = (\<forall><:\<up>\<^sub>\<tau> n k T. \<up>\<^sub>\<tau> n (k + 1) U)"
keys_monic: fixes p :: "'a \<Rightarrow>\<^sub>0 'c" shows "keys (punit.monic p) = keys p"
compose_simps_2: fixes ss :: "('c \<Rightarrow> ('d, 'c) Term.term) list" shows "compose (Var # ss) = compose ss"
cod_char: fixes f :: "'a" shows "Right_a.cod f = (if Right_a.arr f then cod f else null)"
clop_def: fixes f :: "'b \<Rightarrow> 'b" shows "clop f = (id \<le> f \<and> mono f \<and> f \<circ> f \<le> f)"
lcg_through_make_step: fixes Rs :: "(('b, 'c) labeled_graph \<times> ('b, 'c) labeled_graph) set"   and init :: "nat \<times> ('b \<times> nat \<times> nat) set"   and sel :: "nat \<times> ('b \<times> nat \<times> nat) set \<Rightarrow> ((('b, 'c) labeled_graph \<times> ('b, 'c) labeled_graph) \<times> ('c \<times> nat) set) option"   and t :: "'d itself" assumes "finite Rs"   and "set_of_graph_rules Rs"   and "graph (graph_of init)"   and "valid_selector Rs sel" shows "least_consequence_graph t Rs (graph_of init) (the_lcg sel Rs init)"
interior_limit_point: fixes x :: "'a"   and S :: "'a set" assumes "x \<in> interior S" shows "x islimpt S"
zip_rev: fixes xs :: "'g list"   and ys :: "'h list" assumes "length xs = length ys" shows "zip (rev xs) (rev ys) = rev (zip xs ys)"
mult_sup_distrib: fixes x :: "V"   and y :: "V"   and z :: "V" shows "x * (y \<squnion> z) = x * y \<squnion> x * z"
value_inv_4: fixes v :: "Syntax.term" shows "\<not> value (Prj1 v)"
h_def: shows "h \<equiv> \<lambda>x. p (SOME y. y \<in> T \<and> open_segment x y \<inter> T = {})"
step: fixes i :: "nat" shows "\<Gamma>\<turnstile> f i \<rightarrow> f (i + 1)"
success_shiftr: fixes h :: "heap"   and i :: "nat ref"   and a :: "'b array"   and key :: "'b" assumes "Ref.get h i < Array.length h a" shows "success (while (shiftr_p a key i) (shiftr_f a i)) h"
geq: fixes w :: "complex" assumes "w \<in> ball z0 r" shows "g w = (w - z0) ^ m * h w"
unit_simps_4: fixes a :: "'c" assumes "C.obj a" shows "D.dom (G.unit a) = G.map\<^sub>0 a"