hcju/isabelleps huggingface dataset:
measurable_on_spike: fixes f :: "'a \<Rightarrow> 'b"   and T :: "'a set"   and S :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "f measurable_on T"   and "negligible S"   and "\<And>x. x \<in> T - S \<Longrightarrow> g x = f x" shows "g measurable_on T"
rchild_def: fixes tree :: "'b tree" shows "rchild tree = (case tree of Leaf \<Rightarrow> Leaf | Node x21 x22 x23 x24 x25 \<Rightarrow> x25)"
max_less_iff_conj: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d"   and z :: "'d" assumes "class.linorder less_eq less" shows "less (ord.max less_eq x y) z = (less x z \<and> less y z)"
distinct_add_tvarsT': fixes acc :: "(variable \<times> String.literal set) list"   and T :: "typ" assumes "distinct acc" shows "distinct (add_tvarsT' T acc)"
FSwSbImorph_termMOD: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'g"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'h"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'g, 'h, 'i) model_scheme" shows "FSwSbImorph h hA termMOD MOD = termFSwSbImorph h hA MOD"
set_cases_3: fixes e :: "'a"   and a :: "('b, 'c, 'a) sum3" assumes "e \<in> set3_sum3 a" obtains "a = In3 e"
sets_sup_measure_F: fixes I :: "'b measure set"   and M :: "'b measure" assumes "finite I"   and "I \<noteq> {}"   and "\<And>i. i \<in> I \<Longrightarrow> sets i = sets M" shows "sets (comm_monoid_set.F sup bot id I) = sets M"
all_dnodes_def: fixes w :: "'b \<times> 'b \<Rightarrow> enat"   and s :: "'b" shows "Dijkstra.all_dnodes w s \<equiv> insert s {v. \<exists>u. w (u, v) \<noteq> \<infinity>}"
fun_var_def: fixes x :: "nat" shows "fun_var x = (\<lambda>x'. if x' = x then 1::'a else (0::'a))"
B0dec: shows "set_nn_integral M (B 0) f = (\<Sum>n. d 0 n)"
clct: shows "y \<succeq>[\<R>] x \<and> independent_vnm (lotteries_on outcomes) \<R> \<and> y \<in> lotteries_on outcomes \<and> x \<in> lotteries_on outcomes \<and> r \<in> lotteries_on outcomes"
Cons_1: assumes "xs \<noteq> []" shows "min_list_param rel xs \<in> set xs"
ws_cdecl_Main: shows "ws_cdecl tprg Main Object"
monotone_gfp_eadd2: fixes x :: "enat" shows "monotone (\<lambda>x y. y \<le> x) (\<lambda>x y. y \<le> x) ((+) x)"
restrict_apply': fixes x :: "'f"   and A :: "'f set"   and f :: "'f \<Rightarrow> 'g" assumes "x \<in> A" shows "restrict f A x = f x"
\<alpha>_dim_2: shows "dim_vec \<alpha> = N"
ideD_3: fixes a :: "'a" assumes "C.ide a" shows "C.cod a = a"
list_induct_CCons: fixes P :: "'b list \<Rightarrow> bool"   and xs :: "'b list" assumes "P []"   and "\<And>x. P [x]"   and "\<And>x y xs. P (y # xs) \<Longrightarrow> P (x # y # xs)" shows "P xs"
remove1_mset_eq_add_mset_iff: fixes A :: "'a multiset"   and a :: "'a"   and A' :: "'a multiset" shows "(remove1_mset a A = add_mset a A') = (A = add_mset a (add_mset a A'))"
degree_sum_smaller: fixes n :: "nat"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'c poly" assumes "0 < n"   and "finite A"   and "\<And>x. x \<in> A \<Longrightarrow> degree (f x) < n" shows "degree (sum f A) < n"
"25_1": assumes "odd n" shows "(\<Sum>k = 1..n - 1. cmod (f (int k))) \<le> 2 * (\<Sum>k = 1..(n - 1) div 2. cmod (f (int k)))"
holds_aand: fixes P :: "'b \<Rightarrow> bool"   and steps :: "'b stream"   and Q :: "'b \<Rightarrow> bool" shows "(holds P steps \<and> holds Q steps) = holds (\<lambda>step. P step \<and> Q step) steps"
simps: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a1 :: "'a list"   and a2 :: "'b list" shows "listrelp r a1 a2 = (a1 = [] \<and> a2 = [] \<or> (\<exists>x y xs ys. a1 = x # xs \<and> a2 = y # ys \<and> r x y \<and> listrelp r xs ys))"
Hfin: fixes x :: "'a" assumes "x \<in> U - S" shows "\<exists>V. open V \<and> x \<in> V \<and> finite {U \<in> \<C>. \<exists>x\<in>V. H U x \<noteq> 0}"
has_products_def: fixes I :: "'a set" shows "J.has_products I \<equiv> I \<noteq> UNIV \<and> (\<forall>J D. discrete_diagram J (\<cdot>\<^sub>J) D \<and> Collect (partial_magma.arr J) = I \<longrightarrow> (\<exists>a. J.has_as_product J D a))"
IH: fixes r :: "nat" assumes "valid_cert xs"   and "1 < p"   and "prod_list ys = r"   and "r \<noteq> 0"   and "\<forall>q\<in>set ys. Prime q \<in> set xs"   and "\<forall>q\<in>set ys. [a ^ ((p - 1) div q) \<noteq> 1] (mod p)" shows "valid_cert (build_fpc p a r ys @ xs)"
lprefix_conv_lappend: fixes xs :: "'b llist"   and ys :: "'b llist" shows "lprefix xs ys = (\<exists>zs. ys = lappend xs zs)"
x_def_2: shows "[[a\<^sub>0 x a\<^sub>n]]"
assms: fixes x :: "'a" assumes "x \<in> set l" shows "P x \<le> (spec r. r = Q x)"
couSim: shows "weak_reduction_coupled_simulation (indRelLTPO TRel) (STCal Source Target)"
shiftl_shiftl: fixes a :: "'a word"   and b :: "nat"   and c :: "nat" shows "a << b << c = a << b + c"
nsum_suc: fixes R :: "('e, 'f) aGroup_scheme"   and f :: "nat \<Rightarrow> 'e"   and n :: "nat" shows "\<Sigma>\<^sub>e R f Suc n = \<Sigma>\<^sub>e R f n \<plusminus>\<^bsub>R\<^esub> f (Suc n)"
filter_zip_map_dup_const: fixes c :: "'c"   and f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "filter (\<lambda>(a, b). a \<noteq> c) (zip (map_dup f (\<lambda>_. c) xs) xs) = filter (\<lambda>(a, b). a \<noteq> c) (zip (map f (remdups xs)) (remdups xs))"
path_connected_component: fixes S :: "'b set" shows "path_connected S = (\<forall>x\<in>S. \<forall>y\<in>S. path_component S x y)"
rank: fixes x :: "(real, 'm) vec" assumes "x \<in> S" shows "rank (matrix (f' x)) < CARD('n)"
of_fun_nth: fixes i :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'c" assumes "i < n" shows "IArray.of_fun f n !! i = f i"
less_cSup_iff: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and X :: "'c set"   and y :: "'c" assumes "class.conditionally_complete_linorder Inf Sup inf less_eq less sup"   and "X \<noteq> {}"   and "preorder.bdd_above less_eq X" shows "less y (Sup X) = (\<exists>x\<in>X. less y x)"
inject_1: fixes x1 :: "'a"   and y1 :: "'a" shows "(L_action.Act x1 = L_action.Act y1) = (x1 = y1)"
sqrt_power2_le: fixes n :: "nat" shows "(local.sqrt n)\<^sup>2 \<le> n"
mpt_factorE_3: fixes proj :: "'a \<Rightarrow> 'd"   and M2 :: "'d measure"   and T2 :: "'d \<Rightarrow> 'd" assumes "I.mpt_factor proj M2 T2" shows "mpt M2 T2"
successive_end_2': fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list"   and x :: "'b"   and y :: "'b" assumes "successive P (xs @ [x])"   and "\<not> P x" shows "successive P (xs @ [x, y])"
simps: fixes base :: "'b"   and elements :: "'a list"   and n :: "nat"   and head :: "'a"   and tail :: "'a iarray"   and prevs :: "(nat \<times> 'b) list" shows "invariant base elements n (head, tail, prevs) = (if elements = [] then prevs = [] else head = hd elements \<and> tail = IArray (tl elements) \<and> set prevs = set_prevs base (tl elements) n)"
SUP_bot_conv_1: fixes Inf :: "'h set \<Rightarrow> 'h"   and Sup :: "'h set \<Rightarrow> 'h"   and inf :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and less_eq :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and less :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and sup :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and bot :: "'h"   and top :: "'h"   and B :: "'i \<Rightarrow> 'h"   and A :: "'i set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "(Sup (B ` A) = bot) = (\<forall>x\<in>A. B x = bot)"
essential_classD3: fixes C :: "'s set"   and x :: "'s"   and y :: "'s" assumes "essential_class C"   and "x \<in> C"   and "y \<in> C" shows "(x, y) \<in> communicating"
id_matrix_dim_1: fixes nr :: "nat"   and nc :: "nat" shows "dim_row (id_matrix nr nc) = nr"
keysFor_insert_MPair: fixes X :: "msg"   and Y :: "msg"   and H :: "msg set" shows "keysFor (insert \<lbrace>X, Y\<rbrace> H) = keysFor H"
RELATES_def: fixes R :: "('b \<times> 'c) set" shows "RELATES R \<equiv> True"
leg1_commutes: shows "src_\<mu>.cod.leg1 \<cdot> src_\<mu>.chine = src_\<mu>.dom.leg1"
bin_nth_rsplit: fixes n :: "nat"   and m :: "nat"   and sw :: "int list"   and nw :: "nat"   and w :: "int"   and k :: "nat" assumes "0 < n"   and "m < n"   and "rev sw = bin_rsplit n (nw, w)"   and "k < length sw" shows "bit (sw ! k) m = bit w (k * n + m)"
bij_betw_index: fixes xs :: "'c list"   and X :: "'c set"   and l :: "nat" assumes "distinct xs"   and "X = set xs"   and "l = length xs" shows "bij_betw (index xs) X {0..<l}"
that: fixes ct :: "'s \<Rightarrow> 's pmf" assumes "ct \<in> Pi\<^sub>E S K"   and "p = v \<circ> simple ct" shows "thesis"
distinct_15: fixes x2 :: "('e, 'f) Term.term"   and x41 :: "('e, 'f) Term.term"   and x42 :: "('e, 'f) Term.term" shows "receive\<langle>x2\<rangle> \<noteq> insert\<langle>x41,x42\<rangle>"
isSng_correct: fixes s :: "'s" assumes "invar s" shows "isSng s = (\<exists>k v. \<alpha> s = [k \<mapsto> v])"
branch_def: fixes subs :: "'a \<Rightarrow> 'a set"   and Gamma :: "'a"   and f :: "nat \<Rightarrow> 'a" shows "branch subs Gamma f = (f 0 = Gamma \<and> (\<forall>n. terminal subs (f n) \<longrightarrow> f (Suc n) = f n) \<and> (\<forall>n. \<not> terminal subs (f n) \<longrightarrow> f (Suc n) \<in> subs (f n)))"
of_rat_eq_1_iff: fixes a :: "rat" shows "(of_rat a = (1::'a)) = (a = 1)"
sample_vars_def: fixes X :: "'v \<Rightarrow> int pmf"   and A :: "'v set"   and m :: "'m" shows "sample_vars X A m = Finite_Set.fold (\<lambda>x m. bind (memo (sample_var X) x) (\<lambda>_. m)) m A"
reduce_shift: fixes f :: "nat \<Rightarrow> real"   and c :: "real" shows "f ` (PR - (kmax f PR \<union> kmin f PR)) = f ` (PR - (kmax (\<lambda>p. f p + c) PR \<union> kmin (\<lambda>p. f p + c) PR))"
psi_def: shows "psi = (\<lambda>f k i x. if x = k then Some i else f x)"
simps: fixes n :: "char list"   and args :: "object list" shows "resolve_instantiate (PAction n args) = instantiate_action_schema (the (resolve_action_schema n)) args"
ide_cod: fixes f :: "'c \<times> 'c \<times> 'c" assumes "C.VVV.arr f" shows "C.VVV.ide (C.VVV.cod f)"
small: shows "small {[a, b]\<^sub>\<circ> |a b. [a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> \<real>\<^sub>\<circ> ^\<^sub>\<times> 2\<^sub>\<nat>}"
swapEnvDom_preserves_wls: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and rho :: "'d \<Rightarrow> 'a \<Rightarrow> ('b, 'c, 'd, 'a, 'f) qTerm set option"   and xs :: "'d"   and x :: "'a"   and y :: "'a" assumes "FixSyn TYPE('a) Delta"   and "FixSyn.wlsEnv Delta rho" shows "FixSyn.wlsEnv Delta (swapEnvDom xs x y rho)"
WTrtFAss: fixes P :: "'addr J_prog"   and h :: "'heap"   and E :: "String.literal \<Rightarrow> ty option"   and e1 :: "'addr expr"   and U :: "ty"   and C :: "String.literal"   and F :: "String.literal"   and T :: "ty"   and fm :: "fmod"   and D :: "String.literal"   and e2 :: "'addr expr"   and T2 :: "ty" assumes "P,E,h \<turnstile> e1 : U"   and "class_type_of' U = \<lfloor>C\<rfloor>"   and "P \<turnstile> C has F:T (fm) in D"   and "P,E,h \<turnstile> e2 : T2"   and "P \<turnstile> T2 \<le> T" shows "P,E,h \<turnstile> e1\<bullet>F{D} := e2 : Void"
local_lipschitz_mult: shows "local_lipschitz UNIV UNIV (*)"
mm_equiv_step: fixes cms\<^sub>1 :: "'Com \<times> (Mode \<Rightarrow> 'Var set)"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and cms\<^sub>2 :: "'Com \<times> (Mode \<Rightarrow> 'Var set)"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and cms\<^sub>1' :: "'Com \<times> (Mode \<Rightarrow> 'Var set)"   and mem\<^sub>1' :: "'Var \<Rightarrow> 'Val" assumes "(cms\<^sub>1, mem\<^sub>1) \<approx> (cms\<^sub>2, mem\<^sub>2)"   and "snd cms\<^sub>1 = snd cms\<^sub>2"   and "(cms\<^sub>1, mem\<^sub>1) \<leadsto> (cms\<^sub>1', mem\<^sub>1')" shows "\<exists>c\<^sub>2' mem\<^sub>2'. (cms\<^sub>2, mem\<^sub>2) \<leadsto> \<langle>c\<^sub>2', snd cms\<^sub>1', mem\<^sub>2'\<rangle> \<and> (cms\<^sub>1', mem\<^sub>1') \<approx> \<langle>c\<^sub>2', snd cms\<^sub>1', mem\<^sub>2'\<rangle>"
assms_2: fixes y :: "variable \<times> String.literal set" assumes "y \<in> tvsT_Set (snd ` set (xs @ [x]))" shows "y \<notin> fst ` set (xs @ [x])"
isSimplePath_fwd: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and s :: "nat"   and p :: "(nat \<times> nat) list"   and t :: "nat" shows "Graph.isSimplePath c s p t = (Graph.isPath c s p t \<and> distinct (pathVertices_fwd s p))"
group_hom_def: fixes G :: "('b, 'c) monoid_scheme"   and H :: "('d, 'e) monoid_scheme"   and h :: "'b \<Rightarrow> 'd" shows "group_hom G H h \<equiv> Group.group G \<and> Group.group H \<and> group_hom_axioms G H h"
iso_arcs_inv_iso: fixes G :: "('e, 'f) pre_digraph"   and hom :: "('e, 'f, 'g, 'h) digraph_isomorphism"   and a :: "'f" assumes "pre_digraph.digraph_isomorphism G hom"   and "a \<in> arcs G" shows "iso_arcs (pre_digraph.inv_iso G hom) (iso_arcs hom a) = a"
SUP_constant: fixes c :: "'al"   and U\<^sub>2 :: "'c set"   and f :: "'c \<Rightarrow> 'al"   and A :: "'c set" assumes "c \<in> U"   and "\<forall>x\<in>U\<^sub>2. f x = c"   and "A \<subseteq> U\<^sub>2" shows "\<Squnion>\<^sub>o\<^sub>w (f ` A) = (if A = {} then \<bottom>\<^sub>o\<^sub>w else c)"
cod_def: fixes f :: "'a" shows "H.cod f = (if H.codomains f \<noteq> {} then SOME b. b \<in> H.codomains f else H.null)"
equiv_vert_uconnected: fixes E' :: "'a uprod set" shows "equiv verts (Restr (uconnected E') verts)"
non_zero_i_eA: shows "\<not> is_zero_row i (echelon_form_of A bezout)"
convex_affine_rel_interior_Int: fixes S :: "'b set"   and T :: "'b set" assumes "convex S"   and "affine T"   and "rel_interior S \<inter> T \<noteq> {}" shows "rel_interior (S \<inter> T) = rel_interior S \<inter> T"
mult_left_one: fixes x :: "'a" shows "top \<sqinter> x = x"
safety_prop_def: fixes X :: "'c program set" shows "safety_prop X = (\<bottom> \<in> X \<and> (\<forall>G. Acts G \<subseteq> \<Union> (Acts ` X) \<longrightarrow> G \<in> X))"
"3_5": shows "poly (hd (p # q # r # ps)) x\<^sub>0 \<noteq> 0"
lossless_S2: fixes \<sigma> :: "bool"   and out2 :: "'grp" shows "lossless_spmf (S2 \<sigma> out2)"
CSS_def: shows "\<lambda>c. fst (snd c) \<equiv> \<lambda>c. fst (snd c)"
Legendre_2_M: shows "Legendre 2 (int M) = 1"
rangeOutside: fixes f :: "('c \<times> 'd) set"   and X :: "'c set" assumes "runiq f"   and "runiq (f\<inverse>)" shows "Range (Outside' X f) = Range f - f `` X"
vector_matrix_mul_assoc: fixes v :: "('b, 'e) vec"   and M :: "(('b, 'd) vec, 'e) vec"   and N :: "(('b, 'c) vec, 'd) vec" shows "v v* M v* N = v v* (M ** N)"
Val\<^sub>1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and v :: "val"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list" shows "P \<turnstile>\<^sub>1 \<langle>Val v,s\<rangle> \<Rightarrow> \<langle>Val v,s\<rangle>"
map\<^sub>0_in_hom_1: fixes a :: "'a" assumes "C.obj a" shows "\<guillemotleft>F.map\<^sub>0 a : F.map\<^sub>0 a \<rightarrow>\<^sub>D F.map\<^sub>0 a\<guillemotright>"
distinct_1: fixes x2 :: "'a" shows "Err \<noteq> OK x2"
clo: shows "closed ((\<lambda>x. (x, f x)) ` S)"
bet_in_angle_bet: fixes A :: "'p"   and B :: "'p"   and P :: "'p"   and C :: "'p" assumes "Bet A B P"   and "P InAngle A B C" shows "Bet A B C"
l_set_child_nodes_get_child_nodes_axioms: shows "l_set_child_nodes_get_child_nodes DocumentClass.type_wf DocumentClass.known_ptr get_child_nodes get_child_nodes_locs set_child_nodes set_child_nodes_locs"
dHyp: shows "0 < \<delta> \<and> (\<forall>\<tau>\<in>T. \<bar>\<tau> - t\<bar> < \<delta> \<longrightarrow> \<parallel>A \<tau> - A t\<parallel>\<^sub>o\<^sub>p \<le> e / (\<parallel>(if s = 0 then 1 else s)\<parallel>))"
fraction_denormal_of_Float: fixes x :: "Float.float" assumes "is_denormal_Float TYPE(('a, 'b) IEEE.float) x" shows "fraction (denormal_of_Float x) = nat \<bar>mantissa x\<bar> * 2 ^ nat (Float.exponent x + int (bias TYPE(('a, 'b) IEEE.float)) + int LENGTH('b) - 1)"
c_snd_less_arg: fixes x :: "nat" assumes "1 < x" shows "c_snd x < x"
span_in_category_axioms: shows "span_in_category (\<odot>) (Dom (SPN \<mu>))"
mheads_cases: fixes emh :: "ref_ty \<times> mhead"   and G :: "prog"   and S :: "qtname"   and t :: "ref_ty"   and sig :: "sig" assumes "emh \<in> mheads G S t sig"   and "wf_prog G" obtains C :: "qtname" and D :: "qtname" and m :: "qtname \<times> methd" where "t = ClassT C" and "declrefT emh = ClassT D" and "accmethd G S C sig = Some m__" and "declclass m__ = D" and "mhead (mthd m__) = mhd emh"   | I :: "qtname" and im :: "qtname \<times> mhead" where "t = IfaceT I__" and "im \<in> accimethds G (pid S) I__ sig" and "mthd im = mhd emh"   | I :: "qtname" and m :: "qtname \<times> methd" where "t = IfaceT I" and "G \<turnstile> Iface I accessible_in pid S" and "accmethd G S Object sig = Some ma__" and "accmodi ma__ \<noteq> Private" and "declrefT emh = ClassT Object" and "mhead (mthd ma__) = mhd emh"   | T :: "ty" and m :: "qtname \<times> methd" where "t = ArrayT T" and "G \<turnstile> T.[] accessible_in pid S" and "accmethd G S Object sig = Some m" and "accmodi m \<noteq> Private" and "declrefT emh = ClassT Object" and "mhead (mthd m) = mhd emh"
conj_ac_2: fixes a :: "'b"   and b :: "'b" shows "a AND b = b AND a"
sd_power_prop_aux: fixes x :: "'a"   and y :: "'b"   and i :: "nat" shows "\<alpha> (x ^ 0) y \<squnion> \<Squnion> {\<alpha> (x ^ k) y |k. 0 < k \<and> k \<le> Suc i} = \<Squnion> {\<alpha> (x ^ k) y |k. k \<le> Suc i}"
insert_Id_image_Acts: fixes f :: "('c \<times> 'c) set \<Rightarrow> ('b \<times> 'b) set"   and F :: "'c program" assumes "f Id = Id" shows "insert Id (f ` Acts F) = f ` Acts F"
a_get_child_nodes_def: fixes ptr :: "(_) object_ptr" shows "a_get_child_nodes ptr = invoke a_get_child_nodes_tups ptr ()"
assms: shows "\<exists>n'. latestAct_cond nid t n n'"
s_ce_1_1_th: fixes x :: "nat"   and y :: "nat"   and e :: "nat" shows "(c_pair x y \<in> nat_to_ce_set e) = (y \<in> nat_to_ce_set (s_ce e x))"
unit_closed: fixes M :: "'b set"   and composition :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit :: "'b" assumes "Group_Theory.monoid M composition unit" shows "unit \<in> M"
LSM_submodule: fixes M :: "('e, 'a, 'f) Module_scheme"   and T :: "'e set" assumes "R module M"   and "T \<subseteq> carrier M" shows "submodule R M (LSM\<^bsub>R\<^esub> M T)"
f': fixes x :: "nat" shows "\<forall>z\<in>S. (f x has_derivative f' x z) (at z within S) \<and> (\<forall>h. norm (f' x z h - g' z h) \<le> inverse (real (Suc x)) * norm h)"
f_pos: shows "\<forall>\<^sub>F x in sequentially. 0 < f x"
rows_def: fixes A :: "'a mat" shows "rows A = map (row A) [0..<dim_row A]"
EM_remainder_converges_def: fixes n :: "nat"   and f :: "real \<Rightarrow> 'b"   and a :: "int" shows "EM_remainder_converges n f a = (\<exists>L. ((\<lambda>x. EM_remainder' n f (real_of_int a) (real_of_int x)) \<longlongrightarrow> L) at_top)"
indep_setsD: fixes F :: "'d \<Rightarrow> 'a set set"   and I :: "'d set"   and J :: "'d set"   and A :: "'d \<Rightarrow> 'a set" assumes "indep_sets F I"   and "J \<subseteq> I"   and "J \<noteq> {}"   and "finite J"   and "\<forall>j\<in>J. A j \<in> F j" shows "prob (\<Inter> (A ` J)) = (\<Prod>j\<in>J. prob (A j))"
induces_meta_adjunction: shows "meta_adjunction (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) Adj.G.F G Adj.G.\<phi> Adj.G.\<psi>"
rep_monomial: fixes x :: "'b"   and c :: "'a" shows "rep (monomial c x) = c *s x"
sign_Min_lt_0: fixes bin :: "int" shows "(bin_sign bin = - 1) = (bin < 0)"
finite_has_maximal2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and a :: "'b" assumes "class.order less_eq less"   and "finite A"   and "a \<in> A" shows "\<exists>m\<in>A. less_eq a m \<and> (\<forall>b\<in>A. less_eq m b \<longrightarrow> m = b)"
surjective: fixes r :: "('a, 'b) graph_scheme" shows "r = \<lparr>nodes = nodes r, edges = edges r, \<dots> = more r\<rparr>"
div_mod_decomp_int: fixes A :: "int"   and n :: "int" shows "A = A div n * n + A mod n"
le_imp_inverse_le: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq a b"   and "less zero a" shows "less_eq (inverse b) (inverse a)"
unsat_indices_def: fixes dir :: "('a, 'b) Direction"   and s :: "('a, 'b) state"   and vs :: "nat list"   and eq :: "nat \<times> linear_poly" shows "unsat_indices dir s vs eq = (let r = rhs eq; li = Direction.LI dir s; ui = Direction.UI dir s in remdups (li (lhs eq) # map (\<lambda>x. if coeff r x < 0 then li x else ui x) vs))"
that: fixes n :: "nat" assumes "\<And>m. n \<le> m \<Longrightarrow> bit k m = bit k n"   and "0 < n \<Longrightarrow> bit k (n - 1) \<noteq> bit k n" shows "thesis"
algebraic_altdef: fixes x :: "'a" shows "algebraic x = (\<exists>p. (\<forall>i. poly.coeff p i \<in> \<rat>) \<and> p \<noteq> 0 \<and> poly p x = (0::'a))"
Uhyp_1: fixes s :: "'a" assumes "s \<in> S" shows "is_interval (U s)"
Sup_exp_def: fixes S :: "('b \<Rightarrow> real) set" shows "Sup_exp S = (if S = {} then \<lambda>s. 0 else (\<lambda>s. Sup {f s |f. f \<in> S}))"
Cn_equiv: fixes A :: "'a set"   and p :: "'a"   and q :: "'a" shows "(A \<turnstile> p .\<longleftrightarrow>. q) = (Cn (A \<union> {p}) = Cn (A \<union> {q}))"
Units_pow_closed: fixes x :: "'a"   and d :: "nat" assumes "x \<in> Units G" shows "x [^] d \<in> Units G"
mkArr_Fun: fixes f :: "'s" assumes "S.arr f" shows "S.mkArr (S.set (S.dom f)) (S.set (S.cod f)) (S.Fun f) = f"
b_less_1: fixes k :: "nat"   and as :: "real list"   and bs :: "real list"   and b :: "real" assumes "akra_bazzi_params k as bs"   and "b \<in> set bs" shows "b < 1"
simps_2: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and xs :: "'a list"   and vs :: "'b list" shows "remdups_wrt_rev f (x # xs) vs = (let fx = f x in if List.member vs fx then remdups_wrt_rev f xs vs else x # remdups_wrt_rev f xs (fx # vs))"
satC_def: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" shows "TE.satC \<xi> \<equiv> list_ex (TE.satL \<xi>)"
assms_1: shows "rreq_rrep_fresh (rt (\<sigma> sip)) (Rreq hops dip dsn dsk oip osn sip)"
dual_dual_ord: fixes dual :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.ord_with_dual dual less_eq" shows "less_eq (dual x) (dual y) = less_eq y x"
gC_satC: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and c :: "('fsym, 'psym) lit list" assumes "Ik.satC \<xi> c" shows "GE.satC \<xi> (gC c)"
g_def: shows "g \<equiv> Cn 1 r_ifeq_else_diverg [f, Z, Z]"
hs_ins_def: fixes x :: "'a"   and ht :: "('a, unit) hashtable" shows "hs_ins x ht \<equiv> hm_update x () ht"
ne: shows "\<guillemotleft> G \<guillemotright> && (wp a Q \<circ> \<phi>) \<noteq> \<guillemotleft> G \<guillemotright> && wp b (Q \<circ> \<phi>)"
q_cong_reverse: fixes a :: "nat" shows "[q + a = a] (mod q)"
image_affinity_cbox: fixes m :: "real"   and c :: "'a"   and a :: "'a"   and b :: "'a" shows "(\<lambda>x. m *\<^sub>R x + c) ` cbox a b = (if cbox a b = {} then {} else if 0 \<le> m then cbox (m *\<^sub>R a + c) (m *\<^sub>R b + c) else cbox (m *\<^sub>R b + c) (m *\<^sub>R a + c))"
openin_closedin: fixes S :: "'c set"   and U :: "'c topology" assumes "S \<subseteq> topspace U" shows "openin U S = closedin U (topspace U - S)"
Sem_deterministic: fixes c :: "'a com"   and s :: "'a option"   and s1 :: "'a option"   and s2 :: "'a option" assumes "Sem c s s1"   and "Sem c s s2" shows "s1 = s2"
assumption_conjuncive: fixes p :: "'b Assertion" shows "[\<cdot> p ] \<in> conjunctive"
antisym: fixes x :: "'c"   and y :: "'c" assumes "bob x y"   and "bob y x"   and "x \<in> B" shows "x = y"