hcju/isabelleps huggingface dataset:
mult_rat_cancel: fixes c :: "int"   and a :: "int"   and b :: "int" assumes "c \<noteq> 0" shows "Rat.Fract (c * a) (c * b) = Rat.Fract a b"
strict_sorted_appendE2: fixes xs :: "'c list"   and ys :: "'c list"   and x :: "'c" assumes "Lists_Ex.strict_sorted (xs @ ys)"   and "x \<in> set xs" shows "\<forall>y\<in>set ys. x < y"
simps_1: fixes \<S> :: "nat \<Rightarrow> bool option" shows "no_write_to_read_only_memory \<S> [] = True"
empty: fixes g :: "'g" shows "g \<turnstile> [Entry g]\<Down>\<^sub>s ssaStep g (Entry g) 0 Map.empty"
simps_2: fixes \<pi> :: "Preliminaries.var \<Rightarrow> 'e trm"   and p :: "'f"   and Tl :: "'e trm list" shows "substA \<pi> (Pr p Tl) = Pr p (map (subst \<pi>) Tl)"
arestr_lit: fixes v :: "'c"   and a :: "'d \<Longrightarrow> 'e" shows "\<guillemotleft>v\<guillemotright> \<restriction>\<^sub>e a = \<guillemotleft>v\<guillemotright>"
2: shows "trees_cyl ` trees (sets M) \<subseteq> Pow (space (tree_sigma M))"
scale_zero_right: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'b" assumes "module scale" shows "scale a (0::'c) = (0::'c)"
map_comp: fixes g :: "'f \<Rightarrow> 'e"   and f :: "'g \<Rightarrow> 'f"   and v :: "('d, 'g) fmap" shows "fmmap g (fmmap f v) = fmmap (g \<circ> f) v"
Abs_equal_inverse: fixes y :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "y \<in> {(=)}" shows "Rep_equal (Abs_equal y) = y"
measurable: fixes x :: "'a" assumes "x \<in> A" shows "set_borel_measurable M {a..} (f x)"
psubstl1_correct: fixes l :: "'a list"   and mps :: "(mon \<times> pol) list"   and P1 :: "pol"   and n :: "nat" assumes "in_carrier l"   and "Imon_pol_list l mps" shows "Ipol l (psubstl1 P1 mps n) = Ipol l P1"
nat_filtration_not_borel_info': fixes f :: "bool stream \<Rightarrow> 'a"   and n :: "nat"   and N :: "'a measure" assumes "f \<in> nat_filtration n \<rightarrow>\<^sub>M N"   and "set_discriminating n f N" shows "f \<circ> pseudo_proj_False n = f"
thunks_subst: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and y :: "Vars.var"   and x :: "Vars.var" shows "thunks \<Gamma>[y::h=x] = thunks \<Gamma>"
that: fixes D' :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry" assumes "A \<turnstile> \<langle>l, D\<rangle> \<leadsto>\<^sub>\<N>* \<langle>l', D'\<rangle>"   and "[D']\<^bsub>v,n\<^esub> = Z"   and "valid_dbm D'" shows "thesis"
subst_lsst_nil: fixes \<delta> :: "'f \<Rightarrow> ('e, 'f) Term.term" shows "[] \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta> = []"
simps_1: fixes Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" shows "vc SKIP Q = True"
simps_6: fixes v :: "String.literal" shows "ground_type (Class v) = Class v"
shows_prec_int_def: fixes d :: "nat"   and i :: "int" shows "shows_prec d i = (if i < 0 then shows CHR ''-'' \<circ> shows (nat (- i)) else shows (nat i))"
opt_map_def: fixes f :: "'b \<Rightarrow> 'd option"   and g :: "'d \<Rightarrow> 'c option" shows "f |> g \<equiv> \<lambda>s. case f s of None \<Rightarrow> None | Some x \<Rightarrow> g x"
sign_simps_35: fixes am :: "'d"   and bm :: "'d"   and ck :: "'d" shows "(am + bm) * ck = am * ck + bm * ck"
simps_3: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and va :: "('a, 'b) rbt"   and vb :: "'a"   and vc :: "'b"   and vd :: "('a, 'b) rbt"   and y :: "'a"   and s :: "'b"   and b :: "('a, 'b) rbt" shows "ord.rbt_del_from_left less x (Branch R va vb vc vd) y s b = Branch R (ord.rbt_del less x (Branch R va vb vc vd)) y s b"
Sup_inners_Sup_inner: shows "(Sup_inners, Sup_inner) \<in> \<langle>lv_rel\<rangle>set_rel \<rightarrow> lv_rel \<rightarrow> \<langle>rnv_rel\<rangle>nres_rel"
prime_ideal_proper: fixes I :: "'a set" assumes "prime_ideal R I" shows "I \<noteq> carrier R"
ask_state_def: fixes ask :: "('i \<Rightarrow> 'h) \<Rightarrow> 'h"   and f :: "'i \<Rightarrow> ('g, 'h) stateT" shows "ask_state ask f = StateT (\<lambda>s. ask (\<lambda>r. run_state (f r) s))"
simps_7: fixes v :: "'b" shows "(dRETURN v \<le> dSUCCEEDi) = False"
SAMEQUOT_def: fixes a :: "'d"   and A :: "'d list set" shows "SAMEQUOT a A = {fin_cut_same a x @ replicate m a |x m. x \<in> A}"
filterlim_real_at_infinity_sequentially: shows "filterlim real at_infinity sequentially"
llist_inf_eq_LNil: fixes xs :: "'b llist"   and ys :: "'b llist" shows "lnull (xs \<sqinter> ys) = (xs \<noteq> LNil \<longrightarrow> ys \<noteq> LNil \<longrightarrow> lhd xs \<noteq> lhd ys)"
assms_2: fixes x :: "'a" assumes "x \<in> set ys" shows "f x = g x"
minus_eq: fixes x :: "'a"   and y :: "'a" shows "x \<ominus> y = x \<oplus> \<ominus> y"
ind_in_set_assoc: fixes C :: "nat set"   and B :: "nat set"   and A :: "nat set" assumes "C \<subseteq> B"   and "B \<subseteq> A" shows "ind_in_set (ind_in_set A ` B) ` ind_in_set A ` C = ind_in_set B ` C"
eval\<^sub>c_bij: fixes b_of_a :: "'c \<Rightarrow> 'd"   and F :: "char list \<Rightarrow> 'c list \<Rightarrow> 'c"   and G :: "char list \<Rightarrow> 'c list \<Rightarrow> bool"   and C :: "fterm literal set" assumes "bij b_of_a" shows "eval\<^sub>c (F_conv b_of_a F) (G_conv b_of_a G) C = eval\<^sub>c F G C"
induct: fixes x :: "dbtm"   and P :: "dbtm \<Rightarrow> bool" assumes "wf_dbtm x"   and "P DBZero"   and "\<And>name. P (DBVar name)"   and "\<And>t1 t2. \<lbrakk>wf_dbtm t1; P t1; wf_dbtm t2; P t2\<rbrakk> \<Longrightarrow> P (DBEats t1 t2)" shows "P x"
mult_le_cancel_right2: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c"   and c :: "'c" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (times a c) c = ((less zero c \<longrightarrow> less_eq a one) \<and> (less c zero \<longrightarrow> less_eq one a))"
comb_planar_impl_altdef: fixes G :: "'b list \<times> ('b \<times> 'b) list" shows "comb_planar_impl G = (\<exists>M\<in>set (all_maps_list G). genus_impl G M = 0)"
fds_moebius_inversion: fixes f :: "'a fds"   and g :: "'a fds" shows "(f = fds moebius_mu * g) = (g = f * fds_zeta)"
fps_of_poly_nth: fixes p :: "'b poly"   and n :: "nat" shows "fps_of_poly p $ n = coeff p n"
arg_pi2_iff: fixes z :: "complex" shows "(z \<noteq> 0 \<and> arg z = pi / 2) = (is_imag z \<and> 0 < Im z)"
distinct_clinit_def: fixes frs :: "frame list" shows "distinct_clinit frs \<equiv> distinct (clinit_classes frs)"
LitI: fixes G :: "prog"   and s :: "state"   and v :: "val" shows "G\<turnstile>s \<midarrow>Lit v-\<succ>(if normal s then v else undefined)\<rightarrow> s"
cmp_simps_5: fixes f :: "'b"   and g :: "'b" assumes "B.ide f"   and "B.ide g"   and "src\<^sub>B f = trg\<^sub>B g" shows "C.cod (\<Phi>\<^sub>F (f, g)) = F (f \<star>\<^sub>B g)"
alive_alloc_cases: fixes x :: "Value"   and s :: "Store"   and t :: "New" assumes "alive x (s\<langle>t\<rangle>)" obtains "alive x s"   | "x = new s t"
assms_1: shows "constructive_security_obsf real_resource ideal_resource sim \<I>_real \<I>_ideal \<I>_common \<A> adv"
cscc_invar_ext_axioms: shows "cscc_invar_ext G l D0"
omega_nonzero: shows "\<omega> \<noteq> []\<^sub>\<circ>"
distinct_9: fixes x2 :: "int"   and x4 :: "int" shows "Intv' x2 \<noteq> Smaller' x4"
parts_image_priEK: fixes C :: "agent"   and KK :: "nat set"   and evs :: "event list" assumes "Key (priEK C) \<in> parts (Key ` KK \<union> knows Spy evs)"   and "evs \<in> set_cr" shows "priEK C \<in> KK \<or> C \<in> bad"
simps_1: fixes n :: "nat" shows "get_fstn_args n 0 = []"
set_eqI: fixes A :: "'a set"   and B :: "'a set" assumes "\<And>x. (x \<in> A) = (x \<in> B)" shows "A = B"
cone_closed_plus: fixes a :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and hU :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set"   and b :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "a \<in> cone hU"   and "b \<in> cone hU" shows "a + b \<in> cone hU"
r: fixes c :: "'a set set" assumes "c \<in> chains S"   and "\<exists>x. x \<in> c" shows "\<Union> c \<in> S"
equivalence_functor_axioms: shows "equivalence_functor (\<cdot>) (\<cdot>) (C.tensor (C.cod \<iota>))"
read_state_reg_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> get_S (cpu_reg_val PSR s1) = 0 \<and> get_S (cpu_reg_val PSR s2) = 0 \<and> t1 = snd (fst (read_state_reg_instr instr s1)) \<and> t2 = snd (fst (read_state_reg_instr instr s2))" shows "low_equal t1 t2"
dim_eq_span: fixes S :: "'b set"   and T :: "'b set" assumes "S \<subseteq> T"   and "vs1.dim T \<le> vs1.dim S" shows "vs1.span S = vs1.span T"
AbstTermP_imp_OrdP: fixes v :: "tm"   and i :: "tm"   and t :: "tm"   and u :: "tm" shows "{AbstTermP v i t u} \<turnstile> OrdP i"
f2: shows "length (vec_plus (vec A) (vec B)) = length (vec A)"
of_nat_1: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c" assumes "class.semiring_1 one times plus zero" shows "semiring_1.of_nat one plus zero 1 = one"
simps_2: fixes b :: "bool"   and m :: "'b \<Rightarrow> 'b trie option"   and k :: "'b"   and xs :: "'b list" shows "isin (Nd b m) (k # xs) = (case m k of None \<Rightarrow> False | Some t \<Rightarrow> isin t xs)"
r_sub_smult: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set"   and K :: "'e set" assumes "Group G"   and "G \<guillemotright> H "   and "G \<guillemotright> K " shows "K \<subseteq> H \<diamondop>\<^bsub>G\<^esub> K"
append_fun_equiv: fixes t1' :: "(val \<times> val) list"   and t1 :: "(val \<times> val) list"   and t2' :: "(val \<times> val) list"   and t2 :: "(val \<times> val) list" assumes "t1' \<sim> t1"   and "t2' \<sim> t2" shows "t1' @ t2' \<sim> t1 @ t2"
assms_1: fixes b :: "'a" shows "monotone orda ordb (\<lambda>f. tnil f b)"
not_Infinitesimal_not_zero: fixes x :: "'a star" assumes "x \<notin> Infinitesimal" shows "x \<noteq> 0"
isTrue_exi_iff_PP: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "isTrue (PP \<langle>\<phi>\<rangle>) = (\<exists>n\<in>num. isTrue (PPf n \<langle>\<phi>\<rangle>))"
rep_eq: fixes x :: "('d, 'b) tllist"   and xa :: "('d, 'b) tllist" shows "local.tllist_ord x xa = (case (llist_of_tllist x, terminal x) of (xs1, b1) \<Rightarrow> \<lambda>(xs2, b2). if lfinite xs1 then b1 = b \<and> lprefix xs1 xs2 \<or> xs1 = xs2 \<and> flat_ord b b1 b2 else xs1 = xs2) (llist_of_tllist xa, terminal xa)"
remove_term_keys: fixes p :: "'a mpoly"   and m :: "nat \<Rightarrow>\<^sub>0 nat" shows "keys (mapping_of p) - {m} = keys (mapping_of (remove_term m p))"
vsv_is_VLambda: fixes A :: "V"   and f :: "V \<Rightarrow> V" shows "VLambda (\<D>\<^sub>\<circ> (VLambda A f)) (app (VLambda A f)) = VLambda A f"
fmdom'_fmrestrict_set: fixes X :: "'d set"   and f :: "('d, 'e) fmap" shows "fmdom' (fmrestrict_set X f) = X \<inter> fmdom' f"
Plus_OneR: fixes r :: "'a language" assumes "\<oo> r" shows "Plus r One = r"
finite_Sup_less_iff: fixes X :: "'a set"   and a :: "'a" assumes "finite X"   and "X \<noteq> {}" shows "(Sup X < a) = (\<forall>x\<in>X. x < a)"
enumerator_equiv: fixes n :: "nat"   and m :: "nat"   and "is" :: "nat list" assumes "2 < n"   and "1 < m" shows "(is \<in> set (enumerator n m)) = incrIndexList is n m"
UNIV_not_singleton: fixes x :: "'c" shows "UNIV \<noteq> {x}"
max_bag_card_lower_bound_1: assumes "V \<noteq> {}" shows "0 < max_bag_card"
some_trg_eqI: fixes b :: "'a"   and \<mu> :: "'a" assumes "b \<in> targets \<mu>"   and "some_trg b = b" shows "some_trg \<mu> = b"
bv_msb_Cons: fixes b :: "bit"   and bs :: "bit list" shows "bv_msb (b # bs) = b"
hyps: fixes cs :: "('a, 'b, 'c) com list" shows "\<Gamma>\<turnstile>cs,css \<Down> Fault f"
vifunion_iff: fixes b :: "V"   and I :: "V"   and f :: "V \<Rightarrow> V" shows "(b \<in>\<^sub>\<circ> \<Union>\<^sub>\<circ> (VLambda I f `\<^sub>\<circ> I)) = (\<exists>i\<in>elts I. b \<in>\<^sub>\<circ> f i)"
set_insort_key: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and x :: "'b"   and xs :: "'b list" assumes "class.linorder less_eq less" shows "set (linorder.insort_key less_eq f x xs) = insert x (set xs)"
par_Inf_distrib: fixes par :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and skip :: "'b::refinement_lattice"   and D :: "'b::refinement_lattice set"   and c :: "'b::refinement_lattice" assumes "par_distrib (par::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (skip::'b::refinement_lattice)"   and "(D::'b::refinement_lattice set) \<noteq> {}" shows "(par::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (c::'b::refinement_lattice) (\<Sqinter> (D::'b::refinement_lattice set)) = \<Sqinter> (par c ` D)"
assms_1: shows "w\<^sub>1 \<preceq>\<^sub>F\<^sub>I w\<^sub>2"
affine_parallel_reflex: fixes S :: "'a set" shows "affine_parallel S S"
action_mult: fixes G :: "('c, 'd) monoid_scheme"   and \<phi> :: "'c \<Rightarrow> 'e \<Rightarrow> 'e"   and M :: "'e set"   and g :: "'c"   and h :: "'c"   and m :: "'e" assumes "group_action G \<phi> M"   and "g \<in> carrier G"   and "h \<in> carrier G"   and "m \<in> M" shows "\<phi> g (\<phi> h m) = \<phi> (g \<otimes>\<^bsub>G\<^esub> h) m"
deriv_nonneg_imp_mono: fixes a :: "real"   and b :: "real"   and g :: "real \<Rightarrow> real"   and g' :: "real \<Rightarrow> real" assumes "\<And>x. x \<in> {a..b} \<Longrightarrow> (g has_real_derivative g' x) (at x)"   and "\<And>x. x \<in> {a..b} \<Longrightarrow> 0 \<le> g' x"   and "a \<le> b" shows "g a \<le> g b"
cases: fixes a1 :: "('p, 's, 'm) configuration"   and a2 :: "('p, 's, 'm) event list"   and a3 :: "('p, 's, 'm) configuration" assumes "trace a1 a2 a3" obtains c :: "('p, 's, 'm) configuration" where "a1 = ca__" and "a2 = []" and "a3 = ca__"   | c :: "('p, 's, 'm) configuration" and ev :: "('p, 's, 'm) event" and c' :: "('p, 's, 'm) configuration" and t :: "('p, 's, 'm) event list" and c'' :: "('p, 's, 'm) configuration" where "a1 = c" and "a2 = ev # t" and "a3 = c''" and "c \<turnstile> ev \<mapsto> c'" and "trace c' t c''"
real_times_code: fixes x :: "rat"   and y :: "rat" shows "Ratreal x * Ratreal y = Ratreal (x * y)"
trivial_reduced_homology_group_nsphere: fixes p :: "int"   and n :: "nat" shows "trivial_group (reduced_homology_group p (nsphere n)) = (p \<noteq> int n)"
simps: fixes uu :: "'a ext0" shows "uu\<^sup>\<star> = ext0.Top"
nz_2: shows "of_complex v' \<noteq> 0\<^sub>h"
triangle_right: shows "(g' \<star> \<epsilon>') \<cdot> (\<eta>' \<star> g') = g'"
assms_1: shows "(x, y) \<in> r\<^sup>\<down>"
filterlim_norm_at_top_imp_at_infinity: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter" assumes "LIM x F. norm (f x) :> at_top" shows "filterlim f at_infinity F"
integer_eqI: fixes k :: "integer"   and l :: "integer" assumes "int_of_integer k = int_of_integer l" shows "k = l"
to_F_def: fixes Bot_F :: "'a set"   and Inf_F :: "'a inference set"   and Bot_G :: "'b set"   and Q :: "'c set"   and entails_q :: "'c \<Rightarrow> 'b set \<Rightarrow> 'b set \<Rightarrow> bool"   and Inf_G_q :: "'c \<Rightarrow> 'b inference set"   and Red_I_q :: "'c \<Rightarrow> 'b set \<Rightarrow> 'b inference set"   and Red_F_q :: "'c \<Rightarrow> 'b set \<Rightarrow> 'b set"   and \<G>_F_q :: "'c \<Rightarrow> 'a \<Rightarrow> 'b set"   and \<G>_I_q :: "'c \<Rightarrow> 'a inference \<Rightarrow> 'b inference set option"   and Inf_FL :: "('a \<times> 'd) inference set"   and \<iota>\<^sub>F\<^sub>L :: "('a \<times> 'd) inference" assumes "labeled_lifting_intersection Bot_F Inf_F Bot_G Q entails_q Inf_G_q Red_I_q Red_F_q \<G>_F_q \<G>_I_q Inf_FL" shows "labeled_lifting_intersection.to_F \<iota>\<^sub>F\<^sub>L = Infer (map fst (prems_of \<iota>\<^sub>F\<^sub>L)) (fst (concl_of \<iota>\<^sub>F\<^sub>L))"
pw_ref_iff: fixes S :: "'b nres"   and R :: "('b \<times> 'c) set"   and S' :: "'c nres" shows "(S \<le> \<Down> R S') = (nofail S' \<longrightarrow> nofail S \<and> (\<forall>x. inres S x \<longrightarrow> (\<exists>s'. (x, s') \<in> R \<and> inres S' s')))"
fcompose_poly: fixes q2 :: "'b poly"   and x :: "'b"   and p :: "'b poly"   and q1 :: "'b poly" assumes "poly q2 x \<noteq> (0::'b)" shows "poly p (poly q1 x / poly q2 x) = poly (fcompose p q1 q2) x / poly (q2 ^ degree p) x"
orbit_list_impl_conv: fixes y :: "'b"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and acc :: "'b list" assumes "y \<in> orbit f x" shows "orbit_list_impl f y acc x = rev acc @ iterate 0 (funpow_dist1 f x y) f x"
preserves_equivalence_maps: fixes f :: "'b" assumes "C.equivalence_map f" shows "D.equivalence_map (H f)"
assms_3: shows "S \<subseteq> \<Union> \<C>"
valid_failure': fixes A :: "'c \<Rightarrow> ('b \<times> 'c) option"   and \<sigma> :: "'c"   and M :: "'b \<Rightarrow> 'c \<Rightarrow> (bool \<times> 'c) option" assumes "A \<sigma> = None" shows "\<not> (\<sigma> \<Turnstile> bind\<^sub>S\<^sub>E A M)"
finite_perm_lemma: fixes p :: "perm" shows "finite {a. p \<bullet> a \<noteq> a}"
sat_the_update: fixes y :: "nat"   and \<phi> :: "Formula.formula"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and x :: "event_data option list"   and z :: "event_data option"   and i :: "nat" assumes "y \<notin> fv \<phi>" shows "Formula.sat \<sigma> V (map the (x[y := z])) i \<phi> = Formula.sat \<sigma> V (map the x) i \<phi>"
quotient_group_UN: fixes H :: "'g set" shows "quotient_group H = (\<lambda>g. BinOpSetGroup.Abs_G_perm (G // LCoset_rel H) (+) (LCoset_rel H `` {g})) ` G"
grey_def: fixes r :: "'d" shows "grey r = (\<lambda>s. \<exists>x. r \<in> WL x s)"
lcliff_def: fixes r :: "('b \<times> 'b) set"   and a :: "'b"   and b :: "'b"   and xs :: "(accent \<times> 'b) list" shows "local.lcliff r a b xs = ((\<exists>as b' cs a' bs. snd ` set as \<subseteq> under r a \<and> b' = [(Macron, b)] \<and> snd ` set cs \<subseteq> under r a \<and> a' \<in> {[(Acute, a)], []} \<and> snd ` set bs \<subseteq> under r a \<inter> under r b \<and> xs = as @ b' @ cs @ a' @ bs) \<or> (\<exists>cs a' bs. snd ` set cs \<subseteq> under r a \<union> under r b \<and> a' \<in> {[(Acute, a)], []} \<and> snd ` set bs \<subseteq> under r b \<and> xs = cs @ a' @ bs))"
4: shows "integral Dx_pair (partial_vector_derivative (\<lambda>a. F a \<bullet> j) i) = integral (cbox a b) (\<lambda>x. integral (cbox (g2 x) (g1 x)) (\<lambda>y. partial_vector_derivative (\<lambda>a. F a \<bullet> j) i (y, x)))"
swap_typeI_is_typeII: fixes i :: "real \<times> real"   and j :: "real \<times> real"   and C :: "real \<times> real \<Rightarrow> real \<times> real" assumes "R2 i j"   and "typeI_twoCube C" shows "typeII_twoCube (prod.swap \<circ> C \<circ> prod.swap)"
simps_1: fixes x :: "'c"   and y :: "'d" shows "rel_witness_option (Some x, Some y) = Some (x, y)"
lea_out4__lea: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p"   and A' :: "'p"   and C' :: "'p"   and D' :: "'p"   and F' :: "'p" assumes "A B C LeA D E F"   and "B Out A A'"   and "B Out C C'"   and "E Out D D'"   and "E Out F F'" shows "A' B C' LeA D' E F'"
nodes_list_it_correct: fixes g :: "'G" assumes "invar g" shows "set_iterator (nodes_list_it g) (nodes (\<alpha> g))"
set_relcomp_set: fixes xs :: "('b \<times> 'd) list"   and ys :: "('d \<times> 'c) list" shows "set xs O set ys = fold (\<lambda>(x, y). fold (\<lambda>(y', z) A. if y = y' then insert (x, z) A else A) ys) xs {}"
axioms: fixes A :: "V"   and f :: "V" assumes "binop_bij A f" shows "nop_bij A (2\<^sub>\<nat>) f"
fcard_def: shows "fcard \<equiv> map_fun fset id card"
clauseElFormula_def: fixes clause :: "Literal list"   and formula :: "Literal list list" shows "clause el formula \<equiv> clause \<in> set formula"
elims_2: fixes x :: "'a Point"   and xa :: "'a Cone" assumes "onCone x xa" obtains p :: "'a Point" and cone :: "'a Cone" where "x = p" and "xa = cone" and "space2 (vertex cone) p = slope cone * slope cone * time2 (vertex cone) p"
f_iso: shows "\<forall>x y. x \<le> y \<longrightarrow> f x \<le> f y"
Dube_is_GB_cofactor_bound_indets: fixes F :: "(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set" assumes "finite F" shows "is_GB_cofactor_bound F (Dube (Suc (card (\<Union> (indets ` F)))) (maxdeg F))"
pred_stream_def: shows "pred_stream \<equiv> \<lambda>P x. Ball (set_stream x) P"
assms_12: fixes s1 :: "'b"   and s2 :: "'a"   and a :: "'e"   and b :: "'e" assumes "bad1 s1 = bad2 s2"   and "\<not> bad2 s2 \<Longrightarrow> a = b \<and> X s1 s2"   and "bad2 s2 \<Longrightarrow> X_bad s1 s2" shows "R (a, s1) (b, s2)"
ring_times_0_x: fixes R :: "('e, 'f) Ring_scheme"   and x :: "'e" assumes "Ring R"   and "x \<in> carrier R" shows "\<zero>\<^bsub>R\<^esub> \<cdot>\<^sub>r\<^bsub>R\<^esub> x = \<zero>\<^bsub>R\<^esub>"
in_set_spmf_extend_state_oracle: fixes x :: "'a \<times> 'b \<times> 'c"   and "oracle" :: "'c \<Rightarrow> 'd \<Rightarrow> ('a \<times> 'c) spmf"   and s :: "'b \<times> 'c"   and y :: "'d" shows "(x \<in> set_spmf (\<dagger>oracle s y)) = (fst (snd x) = fst s \<and> (fst x, snd (snd x)) \<in> set_spmf (oracle (snd s) y))"
indep_in_subset: fixes \<E> :: "'a set"   and X :: "'a set"   and Y :: "'a set" assumes "\<E> \<subseteq> \<E>"   and "\<E>.indep_in \<E> X"   and "Y \<subseteq> X" shows "\<E>.indep_in \<E> Y"
mat_adjoint_def: fixes A :: "'b mat" shows "adjoint A \<equiv> mat_of_rows (dim_row A) (map conjugate (cols A))"
maxi_less: fixes xs :: "'c list"   and key :: "'c \<Rightarrow> 'd" assumes "0 < length xs" shows "maxi xs key < length xs"
assms_3: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s T"
"1_2": shows "h \<in> existence_ivl0 x0"
h_pos: fixes x :: "real" assumes "1 \<le> x" shows "0 < h x"
v_def: shows "v \<equiv> expr_sem_rf \<rho> e'"
rep_eq: fixes x :: "('a, 'b) alist" shows "lookup x = map_of (impl_of x)"
simps_4: fixes x :: "'b"   and uu :: "PreSimplyTyped.type"   and A :: "'b ptrm" shows "ptrm_fvs (PFn x uu A) = ptrm_fvs A - {x}"
numeral_power_eq_nat_cancel_iff: fixes x :: "num"   and n :: "nat"   and y :: "int" shows "(numeral x ^ n = nat y) = (numeral x ^ n = y)"
flip_conv: fixes R :: "('d \<times> 'c) set" shows "flip ` R = R\<inverse>"
order_refl_classes1: fixes c :: "classes1" shows "c \<le> c"
BP_def: shows "BP = measure_pmf (bernoulli_pmf p)"
sample_var_def: fixes X :: "'v \<Rightarrow> int pmf"   and x :: "'v" shows "sample_var X x = sample (X x) return"
dim: fixes dim :: "nat"   and f :: "nat \<Rightarrow> complex mat"   and n :: "nat" assumes "matrix_seq dim f" shows "f n \<in> carrier_mat dim dim"
inv_upto_def: fixes n :: "nat"   and p :: "'b mat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and A :: "'b mat"   and m :: "nat" shows "inv_upto n p A m \<equiv> \<forall>i j. i < n \<longrightarrow> j < n \<longrightarrow> j < m \<longrightarrow> p A i j"
C_def: shows "C = real (card Roots') * Radius * exp Radius"
mult_of_int_commute: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and x :: "int"   and y :: "'b" assumes "class.ring_1 one times plus zero minus uminus" shows "times (ring_1.of_int one plus zero minus x) y = times y (ring_1.of_int one plus zero minus x)"
inferI: fixes conc :: "formula list"   and prems :: "formula list set"   and rules :: "(formula list \<times> formula list set) set" assumes "(conc, prems) \<in> rules"   and "prems \<in> Pow (deductions rules)" shows "conc \<in> deductions rules"
cases: fixes r :: "'c graph" obtains nodes :: "'c set" and edges :: "('c \<times> 'c) set" where "r = \<lparr>nodes = nodes, edges = edges\<rparr>"
empty_zone_dbm: shows "\<exists>M. ({} = [M]\<^bsub>v,n\<^esub> \<and> dbm_int M n) \<and> normalized M \<and> (\<forall>k\<le>n. M k k \<le> Le 0)"