hcju/isabelleps huggingface dataset:
irc: fixes r :: "('d \<times> 'd) set"   and linear_cf :: "'d set \<Rightarrow> 'd set"   and X :: "'d set" assumes "linear_cf r linear_cf" shows "irc_on X linear_cf"
exhaust: fixes y :: "parity" obtains "y = Even"   | "y = Odd"   | "y = Either"
set_eq_def: fixes S :: "('c, 'd) eq_object_scheme"   and A :: "'c set"   and B :: "'c set" shows "(A {.=}\<^bsub>S\<^esub> B) = ((\<forall>x\<in>A. x .\<in>\<^bsub>S\<^esub> B) \<and> (\<forall>x\<in>B. x .\<in>\<^bsub>S\<^esub> A))"
substr_eff: fixes x :: "'a extended"   and w :: "('a, 'b option) insert_message list"   and l :: "nat"   and u :: "nat" assumes "x \<in> (InString \<circ> I) ` set (substr w l u)"   and "isOK (idx w x)" shows "l < projr (idx w x) \<and> projr (idx w x) < u"
is_prebicategory: shows "prebicategory (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D"
2: shows "\<exists>v. SubRel.ces v1 es1 v subs \<and> SubRel.ces v es2 v2 subs"
lipschitz_cond_affine: fixes t :: "real"   and T :: "real set"   and A :: "real \<Rightarrow> (('b, 'c) vec, 'd) vec"   and x :: "('b, 'c) vec"   and y :: "('b, 'c) vec" assumes "t \<in> T"   and "bdd_above {\<parallel>A t\<parallel>\<^sub>o\<^sub>p |t. t \<in> T}" shows "\<parallel>A t *v x - A t *v y\<parallel> \<le> Sup {\<parallel>A t\<parallel>\<^sub>o\<^sub>p |t. t \<in> T} * (\<parallel>x - y\<parallel>)"
Diff1_foldSetD: fixes A :: "'b set"   and x :: "'b"   and y :: "'c"   and D :: "'c set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and e :: "'c" assumes "(A\<setminus>{x}, y) \<in> foldSetD D f e"   and "x \<in> A"   and "f x y \<in> D" shows "(A, f x y) \<in> foldSetD D f e"
sphere_empty: fixes r :: "real"   and a :: "'b" assumes "r < 0" shows "sphere a r = {}"
times_monomial_left: fixes t :: "'b"   and c :: "'c"   and p :: "'b \<Rightarrow>\<^sub>0 'c" shows "monomial c t * p = MPoly_Type_Class.punit.monom_mult c t p"
sqrt_exists: fixes x :: "int" assumes "0 \<le> x" shows "\<exists>r'. r' * r' \<le> x \<and> x < (r' + 1) * (r' + 1)"
synth_analz_insert_dy_fake_msg: fixes M :: "msg"   and bad :: "agent set"   and IK :: "msg set"   and CH :: "chan set" assumes "M \<in> dy_fake_msg bad IK CH" shows "synth (analz (insert M IK)) \<subseteq> dy_fake_msg bad IK CH"
update_by_fun_gr_eq_update_by_fun: shows "OAlist_update_by_fun_gr_ntm = OAlist_update_by_fun_ntm"
comm_add_vec: fixes v\<^sub>1 :: "'c vec"   and n :: "nat"   and v\<^sub>2 :: "'c vec" assumes "v\<^sub>1 \<in> carrier_vec n"   and "v\<^sub>2 \<in> carrier_vec n" shows "v\<^sub>1 + v\<^sub>2 = v\<^sub>2 + v\<^sub>1"
match_Field_1: fixes Pm :: "'a \<Rightarrow> ('b \<times> 'b) set"   and Pw :: "'b \<Rightarrow> ('a \<times> 'a) set"   and \<mu> :: "('a \<times> 'b) set"   and m :: "'a"   and w :: "'b" assumes "StableMarriage Pm Pw"   and "StableMarriage.match Pm Pw \<mu>"   and "(m, w) \<in> \<mu>" shows "w \<in> Field (Pm m)"
Cons_1: fixes x :: "real list" assumes "x all_in Is" shows "list_ex ((all_in) x) (split_domain split Is)"
map_ident: fixes t :: "'a \<Rightarrow> 'b" shows "(\<lambda>x. x) \<circ> t = t"
old_document: shows "h \<turnstile> get_owner_document (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^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 child) \<rightarrow>\<^sub>r old_document"
WT_sim: shows "\<I>_real1 \<oplus>\<^sub>\<I> \<I>_real2, \<I>_ideal1 \<oplus>\<^sub>\<I> \<I>_ideal2 \<turnstile>\<^sub>C sim \<surd>"
atLeastAtMost_absI: fixes a :: "real"   and x :: "real" assumes "- a \<le> a"   and "\<bar>x\<bar> \<le> \<bar>a\<bar>" shows "x \<in> {- a..a}"
ideD_1: fixes a :: "'s" assumes "ide a" shows "arr a"
dist_clamps_le_dist_args: fixes a :: "'b"   and b :: "'b"   and y :: "'b"   and x :: "'b" shows "dist (clamp a b y) (clamp a b x) \<le> dist y x"
rewrite_negated_primitives_def: fixes disc_sel :: "('b \<Rightarrow> bool) \<times> ('b \<Rightarrow> 'c)"   and C :: "'c \<Rightarrow> 'b"   and negate :: "('c \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b match_expr"   and m :: "'b match_expr" shows "rewrite_negated_primitives disc_sel C negate m \<equiv> let (spts, rst) = primitive_extractor disc_sel m in if getNeg spts = [] then m else MatchAnd (andfold_MatchExp (map (negate C) (getNeg spts))) (MatchAnd (andfold_MatchExp (map (Match \<circ> C) (getPos spts))) rst)"
poly_as_sum_of_monoms: fixes p :: "'c poly" shows "(\<Sum>i\<le>degree p. monom (coeff p i) i) = p"
assms_1: shows "F \<in> \<lbrakk>H\<rbrakk>\<^sub>H \<rightarrow> \<lbrakk>H\<rbrakk>\<^sub>H"
lgeom_proc_take: fixes i :: "nat"   and n :: "nat"   and u :: "'b"   and d :: "'b"   and init :: "'b"   and w :: "bool stream" assumes "i \<le> n" shows "lgeom_proc u d init (stake i w) = lgeom_proc u d init (take i (stake n w))"
that: fixes c :: "real" assumes "\<And>s. 0 < Re s \<Longrightarrow> cmod (stirling_integral n s) \<le> c / Re s ^ (n - 1)" shows "thesis"
UNIV: fixes S :: "'c set set" shows "generate_topology S UNIV"
LNil_lprefix: fixes xs :: "'b llist" shows "lprefix <> xs"
case_bool_if: shows "case_bool \<equiv> \<lambda>x y P. if P then x else y"
mod_2_eq_odd: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "modulo a (numeral.numeral one plus (num.Bit0 num.One)) = zero_neq_one.of_bool one zero (\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a)"
pr_algo_lts'_def: shows "pr_algo_lts' \<equiv> {((xa, x), xd, xc, xb). pr_algo_lts'p xa x xd xc xb}"
BodyCommute: fixes s :: "'a" assumes "P s"   and "b s" shows "f (c s) = c' (f s)"
convex_imp_locally_path_connected: fixes S :: "'b set" assumes "convex S" shows "locally path_connected S"
first: assumes "a \<in> set u"   and "indoc a u" shows "thesis"
range_cblinfun_code_def: fixes A :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a" shows "range_cblinfun_code A = A *\<^sub>S \<top>"
Digraph_def: fixes G :: "('c, 'd) Graph_scheme" shows "Digraph G \<equiv> E\<^bsub>G\<^esub> \<subseteq> V\<^bsub>G\<^esub> \<times> V\<^bsub>G\<^esub>"
high_proc\<^sub>4_correct: fixes s\<^sub>H :: "stmt" assumes "high_proc\<^sub>4 s\<^sub>H" shows "high_proc\<^sub>2 s\<^sub>H"
prems_2: fixes xs :: "'a list" assumes "ls = ls' @ xs"   and "xs \<noteq> []" shows "thesis"
vsv: fixes a :: "V"   and b :: "V"   and c :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> xs"   and "\<langle>a, c\<rangle> \<in>\<^sub>\<circ> xs" shows "b = c"
pos_divideC_le_eq: fixes c :: "complex"   and b :: "'a"   and a :: "'a" assumes "0 < c" shows "(b /\<^sub>C c \<le> a) = (b \<le> c *\<^sub>C a)"
indiff_set_def: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" shows "indiff_set S A = (A \<noteq> {} \<and> (\<forall>x\<in>A. \<forall>y\<in>A. S x y))"
distinct_17: fixes x2 :: "'b"   and x51 :: "'b ptrm"   and x52 :: "'b ptrm" shows "PVar x2 \<noteq> PPair x51 x52"
equal_pair_count_eq_pairwise: fixes ps1 :: "('b \<times> 'b) list"   and ps2 :: "('c \<times> 'c) list" assumes "length ps1 = length ps2"   and "\<And>i. i < length ps2 \<Longrightarrow> equal_pair (ps1 ! i) = equal_pair (ps2 ! i)" shows "equal_pair_count ps1 = equal_pair_count ps2"
two_composable_identity_arrows_of_spans_axioms: shows "two_composable_identity_arrows_of_spans (\<cdot>) prj0 prj1 (trg f) (trg f)"
fmrestrict_agree_monotonous: fixes f :: "('d, 'e) fmap"   and g :: "('d, 'e) fmap"   and X :: "'d set" assumes "agree f g" shows "agree (fmrestrict_set X f) (fmrestrict_set X g)"
to_total_def: fixes \<Gamma> :: "Eg1.var \<Rightarrow> bexp set option" shows "to_total \<Gamma> \<equiv> \<lambda>v. if v \<in> dom \<Gamma> then the (\<Gamma> v) else dma_type v"
domain: fixes T :: "'e \<Rightarrow> 'h \<Rightarrow> bool"   and Ta :: "'f \<Rightarrow> 'g \<Rightarrow> bool" shows "Domainp (pcr_generator T Ta) = (\<lambda>x. \<exists>y. rel_fun T (rel_step Ta T) x y \<and> terminates y)"
ord: fixes a :: "nat"   and n :: "nat" shows "[a ^ ord n a = 1] (mod n)"
slpa_snoc_intra: fixes cs :: "'edge list"   and as :: "'edge list"   and a :: "'edge" assumes "same_level_path_aux cs as"   and "intra_kind (kind a)" shows "same_level_path_aux cs (as @ [a])"
trace_alt_def: fixes r :: "('d \<times> 'c) stream"   and p :: "'c" shows "trace r p = smap snd r"
maintained_holds: fixes G :: "('c, 'b) labeled_graph"   and e\<^sub>L :: "'c allegorical_term"   and e\<^sub>R :: "'c allegorical_term" assumes ":G:\<lbrakk>e\<^sub>L\<rbrakk> \<subseteq> :G:\<lbrakk>e\<^sub>R\<rbrakk>" shows "maintained (transl_rule (e\<^sub>L \<sqsubseteq> e\<^sub>R)) G"
true_lit_def: fixes I :: "'b set"   and L :: "'b literal" shows "(I \<Turnstile>l L) = (if is_pos L then \<lambda>P. P else Not) (atm_of L \<in> I)"
trivial_factor_iso: shows "the_elem \<in> Group.iso (G Mod {\<one>}) G"
intro: fixes get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "get_parent = l_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_parent get_child_nodes"   and "get_parent_locs = l_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_parent_locs get_child_nodes_locs" shows "l_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_child_nodes get_child_nodes_locs get_parent get_parent_locs"
dl_iso_right_invertible: fixes xf :: "('c, 'd) lifted_distrib_lattice_top" shows "dl_iso (dl_iso_inv xf) = xf"
assms_4: shows "K1 \<lhd> G\<lparr>carrier := K\<rparr>"
it_to_it_def: fixes it :: "('c list \<Rightarrow> bool) \<Rightarrow> ('c \<Rightarrow> 'c list \<Rightarrow> 'c list) \<Rightarrow> 'c list \<Rightarrow> 'c list" shows "it_to_it it \<equiv> foldli (it (\<lambda>_. True) (\<lambda>x l. l @ [x]) [])"
local_demonic_init: fixes init :: "'d set"   and p :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and r :: "'d \<Rightarrow> 'c \<Rightarrow> 'a \<Rightarrow> 'b" shows "local_init init (\<lambda>u. {. id (p u) .} \<circ> [: id (\<lambda>x. id (r u x)) :]) = [: z \<leadsto> (u, x) . u \<in> init \<and> z = x :] \<circ> {.u, x. p u x.} \<circ> [: id (\<lambda>(u, x). id (r u x)) :]"
C3l: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and star :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.C3l_algebra plus times one zero less_eq less star"   and "less_eq (times x y) y" shows "less_eq (times (star x) y) y"
mono_NEGn_NEGc_1: fixes B :: "ty" shows "mono (NEGn B \<circ> NEGc B)"
Gcd_greatest: fixes A :: "'a set"   and a :: "'a" assumes "\<And>b. b \<in> A \<Longrightarrow> a dvd b" shows "a dvd Gcd A"
ANNOT_def: fixes x :: "'e"   and a :: "annot" shows "Autoref_Tagging.ANNOT x a \<equiv> x"
order_tendstoI: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and "open" :: "'a set \<Rightarrow> bool"   and y :: "'a"   and f :: "'b \<Rightarrow> 'a"   and F :: "'b filter" assumes "class.order_topology less_eq less open"   and "\<And>a. less a y \<Longrightarrow> \<forall>\<^sub>F x in F. less a (f x)"   and "\<And>a. less y a \<Longrightarrow> \<forall>\<^sub>F x in F. less (f x) a" shows "filterlim f (topological_space.nhds open y) F"
listI: fixes xs :: "'c list"   and n :: "nat"   and A :: "'c set" assumes "length xs = n"   and "set xs \<subseteq> A" shows "xs \<in> list n A"
Rep_sum_inverse: fixes x :: "'a + 'b" shows "Abs_sum (Rep_sum x) = x"
disc: shows "AEeq N (discounted_value r (cls_val_process Mkt pf) matur) (discounted_value r (\<lambda>m. pyf) matur)"
simps_2: fixes x2 :: "'b equation"   and y2 :: "'b equation" shows "(Neg x2 = Neg y2) = (x2 = y2)"
split_conc: fixes xs :: "('a btree \<times> 'a) list"   and p :: "'a"   and ls :: "('a btree \<times> 'a) list"   and rs :: "('a btree \<times> 'a) list" assumes "split xs p = (ls, rs)" shows "xs = ls @ rs"
cases: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool"   and a1 :: "'a"   and a2 :: "'b list"   and a3 :: "'a" assumes "rtrancl3p r a1 a2 a3" obtains a :: "'a" where "a1 = a__" and "a2 = []" and "a3 = a__"   | a :: "'a" and bs :: "'b list" and a' :: "'a" and b :: "'b" and a'' :: "'a" where "a1 = a" and "a2 = bs @ [b]" and "a3 = a''" and "rtrancl3p r a bs a'" and "r a' b a''"
bind_option_cong_code: fixes x :: "'a option"   and y :: "'a option"   and f :: "'a \<Rightarrow> 'b option" assumes "x = y" shows "Option.bind x f = Option.bind y f"
sup_bot_right: fixes x :: "'a" shows "x \<squnion> \<bottom> = x"
simps_3: fixes v :: "'d strand_label"   and vb :: "poscheckvariant"   and vc :: "('e, 'f) Term.term"   and vd :: "('e, 'f) Term.term" shows "dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p (v, \<langle>vb: vc \<doteq> vd\<rangle>) = (v, \<langle>vb: vc \<doteq> vd\<rangle>)"
ring_mat_simps_3: fixes ty :: "'c itself"   and n :: "nat"   and b :: "'d" shows "\<one>\<^bsub>ring_mat ty n b\<^esub> = 1\<^sub>m n"
plus_const_cases: fixes a1 :: "const"   and a2 :: "const" shows "plus_const a1 a2 = (case (a1, a2) of (Const i, Const j) \<Rightarrow> Const (i + j) | (Const i, Any) \<Rightarrow> Any | (Any, b) \<Rightarrow> Any)"
right_cancel: fixes x :: "nat \<Rightarrow> 'a"   and y :: "nat \<Rightarrow> 'a"   and z :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P"   and "y \<in> carrier P"   and "z \<in> carrier P" shows "(y \<oplus>\<^bsub>P\<^esub> x = z \<oplus>\<^bsub>P\<^esub> x) = (y = z)"
sel_1: fixes x1 :: "'a"   and x2 :: "'a" shows "interval.lower (Interval x1 x2) = x1"
concat_eq: fixes xss :: "'a list list"   and yss :: "'a list list" assumes "list_all2 (\<lambda>xs ys. length xs = length ys) xss yss"   and "concat xss = concat yss" shows "xss = yss"
hyps: assumes "xs prefix of j"   and "i \<in> set (added_ids xs e)" shows "Deliver (i, Create i e) \<in> set xs \<or> (\<exists>mo. Deliver (i, Expunge e mo i) \<in> set xs)"
HC_mono: fixes S :: "'b formula set"   and F :: "'b formula"   and T :: "'b formula set" assumes "S \<turnstile>\<^sub>H F"   and "S \<subseteq> T" shows "T \<turnstile>\<^sub>H F"
prod_cases: fixes P :: "'e \<times> 'f \<Rightarrow> bool"   and p :: "'e \<times> 'f" assumes "\<And>a b. P (a, b)" shows "P p"
infinite_iff_card_of_nat: fixes A :: "'a set" shows "infinite A = (|UNIV| \<le>o |A|)"
tv: fixes x :: "'a" assumes "x \<in> S" shows "v x \<subseteq> S \<and> open (t x) \<and> compact (v x) \<and> (\<exists>u. x \<in> u \<and> u \<subseteq> v x \<and> u = S \<inter> t x)"
strong_del_point_sub_sys: fixes p :: "'a" shows "sub_incidence_system (del_point p) (str_del_point_blocks p) \<V> \<B>"
proper_designI: assumes "int (size \<B>\<^sup>D) \<noteq> 0" shows "proper_design bl \<B>\<^sup>D"
simps_13: fixes x2 :: "expr" shows "Expr x2 \<noteq> Skip"
Gamma_times_zeta_integrable: fixes z :: "complex" assumes "1 < Re z" shows "complex_set_integrable lebesgue {0<..} (\<lambda>x. complex_of_real x powr (z - 1) / complex_of_real (exp x - 1))"
Un_lepoll_mono: fixes A :: "'a set"   and C :: "'b set"   and B :: "'a set"   and D :: "'b set" assumes "A \<lesssim> C"   and "B \<lesssim> D"   and "disjnt C D" shows "A \<union> B \<lesssim> C \<union> D"
Set_inv_lemma: fixes \<tau> :: "'d state \<times> 'd state"   and X :: "'d state \<times> 'd state \<Rightarrow> Set('e)" assumes "\<tau> \<Turnstile> \<delta> X" shows "\<forall>x\<in>\<lceil>\<lceil>Rep_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e (X \<tau>)\<rceil>\<rceil>. x \<noteq> \<bottom>"
qFreshAll_igFreshAll_qInitAll: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and ys :: "'varSort"   and y :: "'var"   and qX :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm"   and qA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs" assumes "igFreshClsSTR MOD" shows "(qFresh ys y qX \<longrightarrow> igFresh MOD ys y (qInit MOD qX)) \<and> (qFreshAbs ys y qA \<longrightarrow> igFreshAbs MOD ys y (qInitAbs MOD qA))"
surjective: fixes r :: "('c, 'd) Order_scheme" shows "r = \<lparr>carrier = carrier r, rel = rel r, \<dots> = Order.more r\<rparr>"
simps_1: fixes x2 :: "nat"   and y2 :: "nat" shows "(Suc x2 = Suc y2) = (x2 = y2)"
set_child_nodes\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def: fixes shadow_root_ptr :: "(_) shadow_root_ptr" shows "set_child_nodes\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr = put_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t shadow_root_ptr RShadowRoot.child_nodes_update"
fact_mod: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "fact n mod fact m = (0::'b)"
sim_range_def: fixes M :: "('d, 'e, 'f) KripkeStructure"   and M' :: "('d, 'e, 'g) KripkeStructure"   and f :: "'f \<Rightarrow> 'g" shows "sim_range M M' f \<equiv> worlds M' = f ` worlds M \<and> (\<forall>a. relations M' a \<subseteq> worlds M' \<times> worlds M')"
char_def: fixes M :: "real measure"   and t :: "real" shows "char M t = CLINT x|M. iexp (t * x)"
in_rel_Collect_case_prod_eq: fixes X :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "in_rel {(x, y). X x y} = X"
simps_13: fixes v :: "'e" shows "generalize_instr (ICall v) = ICall v"
ceiling_eq_iff: fixes x :: "'b"   and a :: "int" shows "(\<lceil>x\<rceil> = a) = (of_int a - (1::'b) < x \<and> x \<le> of_int a)"
conjugate_square_eq_0_vec: fixes v :: "'a vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "(inner_prod v v = (0::'a)) = (v = 0\<^sub>v n)"
monic_imp_div_mod_int_poly_degree: fixes u :: "'b poly"   and p :: "'b poly" assumes "monic u" shows "\<exists>q r. p = q * u + r \<and> (r = 0 \<or> degree r < degree u)"
WEAK: fixes G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and c :: "OBJ"   and A :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool"   and H :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set" assumes "G \<rhd> c : A"   and "G \<subseteq> H" shows "H \<rhd> c : A"
prems_2: shows "F \<^bold>\<rightarrow> G, \<Gamma> \<Rightarrow> \<Delta>"
Rep_model: fixes x :: "model" shows "Rep_model x \<in> model"
sigma_finite_measure_axioms: fixes s :: "'a" shows "sigma_finite_measure (DTMC.T s)"
lim_F_at_bot: shows "(F \<longlongrightarrow> 0) at_bot"
of_real_sqrt: fixes x :: "real" assumes "0 \<le> x" shows "complex_of_real (sqrt x) = csqrt (complex_of_real x)"
assms_3: shows "distinct (cols (submatrix A I UNIV))"
sum_of_odds: fixes n :: "nat" shows "(\<Sum>i = 0..<n. 2 * i + 1) = n ^ Suc (Suc 0)"
gamma_o_Top: shows "\<gamma>\<^sub>o \<top> = UNIV"
inv_comp_left_2: fixes g :: "'c \<times> 'c"   and f :: "'c \<times> 'c" assumes "C.VV.iso (C.VV.comp g f)"   and "C.VV.iso g" shows "C.VV.iso f"
ge_fx: fixes x :: "'a" assumes "x \<in> S" shows "f x \<le> real (n - 1) * e"
of_nat_poly: fixes n :: "nat" shows "of_nat n = [:of_nat n:]"
prv_\<phi>G_eqv: shows "prv (eqv \<phi>G (neg (wrepr.PP \<langle>\<phi>G\<rangle>)))"
iso_tuple_UNIV_I: fixes x :: "'b" shows "s2p UNIV x \<equiv> True"
Reals_minus: fixes a :: "'a" assumes "a \<in> \<real>" shows "- a \<in> \<real>"
carrier_coeff_def: fixes P :: "'c multiset \<Rightarrow> 'a" shows "carrier_coeff P = (\<forall>m. P m \<in> carrier R)"
assoc'_simps_3: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "ide f"   and "ide g"   and "ide h"   and "src f = trg g"   and "src g = trg h" shows "trg \<a>\<^sup>-\<^sup>1[f, g, h] = trg f"
dim_col_conjugate: fixes m :: "'a mat" shows "dim_col (conjugate m) = dim_col m"
assms_2: shows "(q, x) \<in> R\<^sup>*"
ex_partGen: fixes J :: "'a set"   and P :: "'a set set"   and j :: "'a" assumes "J \<in> partGen P"   and "j \<in> J" shows "\<exists>I\<in>P. j \<in> I"
incseq_nn_integral: fixes f :: "nat \<Rightarrow> 'c \<Rightarrow> ennreal"   and M :: "'c measure" assumes "incseq f" shows "incseq (\<lambda>i. integral\<^sup>N M (f i))"
simps_2: fixes e :: "('f, 'g, 'h) exp"   and es :: "('f, 'g, 'h) exp list"   and A :: "'f set option" shows "\<D>s (e # es) A = (\<D> e A \<and> \<D>s es (A \<squnion> \<A> e))"
greaterThanAtMost_eq_atLeastAtMost_diff: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.order less_eq less" shows "ord.greaterThanAtMost less_eq less a b = ord.atLeastAtMost less_eq a b - {a}"
mult_mat_limit: fixes B :: "complex mat"   and m :: "nat"   and X :: "nat \<Rightarrow> complex mat"   and A :: "complex mat" assumes "B \<in> carrier_mat m m"   and "limit_mat X A m" shows "limit_mat (\<lambda>k. X k * B) (A * B) m"
ideal_generated_singleton_subset: fixes d :: "'a"   and S :: "'a set" assumes "d \<in> ideal_generated S"   and "finite S" shows "ideal_generated {d} \<subseteq> ideal_generated S"
llexord_LCons_eq: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c llist"   and ys :: "'c llist"   and x :: "'c" assumes "llexord r xs ys" shows "llexord r (LCons x xs) (LCons x ys)"
1: fixes d :: "nat"   and f :: "nat \<Rightarrow> 'a" shows "vec (Suc d) f $ Suc n = vec d (f \<circ> Suc) $ n"
Int_absorb: fixes A :: "'c set" shows "A \<inter> A = A"
wf_effect_inst_alt: fixes eff :: "object ast_effect" shows "wf_effect_inst eff = wf_effect objT eff"
zip_left_commute: fixes xs :: "'b list"   and ys :: "'c list"   and zs :: "'d list" shows "zip xs (zip ys zs) = map2 (\<lambda>y (x, z). (x, y, z)) ys (zip xs zs)"
adjoint_def: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and I :: "interp"   and \<omega> :: "variable \<Rightarrow> real" shows "USubst.adjoint \<sigma> I \<omega> = mkinterp (\<lambda>f. case SConst \<sigma> f of None \<Rightarrow> Consts I f | Some r \<Rightarrow> term_sem I r \<omega>, \<lambda>f. case SFuncs \<sigma> f of None \<Rightarrow> Funcs I f | Some r \<Rightarrow> \<lambda>d. term_sem (repc I dotid d) r \<omega>, \<lambda>p. case SPreds \<sigma> p of None \<Rightarrow> Preds I p | Some r \<Rightarrow> \<lambda>d. \<omega> \<in> fml_sem (repc I dotid d) r, \<lambda>a. case SGames \<sigma> a of None \<Rightarrow> Games I a | Some r \<Rightarrow> game_sem I r)"
that_6: fixes b\<^sub>1 :: "bool"   and g :: "(C \<rightarrow> CValue) \<rightarrow> C \<rightarrow> CValue" assumes "x = B\<cdot>(Discr b\<^sub>1)"   and "y = CFn\<cdot>g" shows "thesis"
simps: fixes a1 :: "nat"   and a2 :: "'b \<Rightarrow> 'b" shows "swapidseq a1 a2 = (a1 = 0 \<and> a2 = id \<or> (\<exists>n p a b. a1 = Suc n \<and> a2 = Fun.swap a b id \<circ> p \<and> swapidseq n p \<and> a \<noteq> b))"
less_action: fixes P :: "('c, 'd) action"   and Q :: "('c, 'd) action" shows "P < Q \<equiv> P \<sqsubseteq> Q \<and> Q \<notsqsubseteq> P"
simps: fixes M :: "('e, 'f, 'g) FSM" shows "observable M = (\<forall>t s1. succ M t s1 = {} \<or> (\<exists>s2. succ M t s1 = {s2}))"
join_1: shows "s' = (s(r \<mapsto> (\<sigma>, \<tau>;;\<tau>', \<E> [VE (CV Unit)])))(r' := None)"
posInfCase: assumes "w = POS_INF" shows "\<exists>r. Rep_bword bw \<equiv>\<^sub>E r"
inter_distr1: fixes j :: "nat_int"   and k :: "nat_int"   and i :: "nat_int" shows "consec j k \<longrightarrow> i \<sqinter> (j \<squnion> k) = i \<sqinter> j \<squnion> i \<sqinter> k"