hcju/isabelleps huggingface dataset:
to_conjugate_space_inverse: fixes y :: "'c" assumes "y \<in> UNIV" shows "from_conjugate_space (to_conjugate_space y) = y"
one_not_zero: fixes R :: "('e, 'f) ring_scheme" assumes "domain R" shows "\<one>\<^bsub>R\<^esub> \<noteq> \<zero>\<^bsub>R\<^esub>"
integrableI_nn_integral_finite: fixes f :: "'b \<Rightarrow> real"   and M :: "'b measure"   and x :: "real" assumes "f \<in> borel_measurable M"   and "AE x in M. 0 \<le> f x"   and "\<integral>\<^sup>+ x. ennreal (f x) \<partial>M = ennreal x" shows "integrable M f"
prj_chine_assoc_3: shows "\<mu>_\<nu>\<pi>_\<rho>.Prj\<^sub>0\<^sub>0 \<cdot> \<mu>_\<nu>\<pi>_\<rho>.chine_assoc = \<mu>_\<nu>\<pi>_\<rho>.Prj\<^sub>0"
sgn_power_injE: fixes a :: "real"   and n :: "nat"   and x :: "real"   and b :: "real" assumes "sgn a * \<bar>a\<bar> ^ n = x"   and "x = sgn b * \<bar>b\<bar> ^ n"   and "0 < n" shows "a = b"
is_glbD1: fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and S :: "'c set"   and x :: "'c" assumes "class.po below"   and "po.is_glb below S x" shows "po.is_lb below S x"
fold_expand: shows "foldl Gauss_Jordan_column_k_det_P (1::'a, 0, A) [0..<Suc k] = (fst (foldl Gauss_Jordan_column_k_det_P (1::'a, 0, A) [0..<Suc k]), fst (snd (foldl Gauss_Jordan_column_k_det_P (1::'a, 0, A) [0..<Suc k])), snd (snd (foldl Gauss_Jordan_column_k_det_P (1::'a, 0, A) [0..<Suc k])))"
striaghten_downtop_computation: shows "kauff_mat (basic [cup, vert] \<circ> basic [vert, cap]) = kauff_mat (basic [vert] \<circ> basic [vert])"
binomial_int_ineq_1: fixes x :: "int"   and y :: "int"   and k :: "nat" assumes "0 \<le> x"   and "0 \<le> y" shows "2 * (x + y gchoose k) \<le> x gchoose k + (x + 2 * y gchoose k)"
WT_converterD': fixes \<I> :: "('a, 'b) \<I>"   and \<I>' :: "('c, 'd) \<I>"   and conv :: "('a, 'b, 'c, 'd) converter"   and q :: "'a" assumes "\<I>, \<I>' \<turnstile>\<^sub>C conv \<surd>"   and "q \<in> outs_\<I> \<I>" shows "\<I>' \<turnstile>g run_converter conv q \<surd> \<and> (\<forall>(r, conv')\<in>results_gpv \<I>' (run_converter conv q). r \<in> responses_\<I> \<I> q \<and> \<I>, \<I>' \<turnstile>\<^sub>C conv' \<surd>)"
intros_5: fixes a :: "'d"   and b :: "'d" assumes "a \<le> b" shows "Lt a \<prec> Le b"
preservation: fixes v :: "val"   and e :: "exp"   and \<rho> :: "(nat \<times> val) list"   and e' :: "exp" assumes "v \<in> E e \<rho>"   and "e \<longrightarrow>* e'" shows "v \<in> E e' \<rho>"
bounded_linear_axioms: fixes x :: "'a" shows "bounded_linear (f' y x)"
key_le_right: fixes c :: "color"   and lt :: "('b, 'c) RBT_Impl.rbt"   and k :: "'b"   and v :: "'c"   and rt :: "('b, 'c) RBT_Impl.rbt"   and x :: "'b" assumes "rbt_sorted (Branch c lt k v rt)"   and "x \<in> set (RBT_Impl.keys rt)" shows "k \<le> x"
asym: fixes a :: "'t"   and b :: "'t" assumes "a \<prec>\<^sub>t b"   and "b \<prec>\<^sub>t a" shows "False"
S2: shows "f (funpower g n I) = funpower h n (f I)"
functor_axioms: shows "functor (\<cdot>\<^sub>J) (\<cdot>) Cod_g.map"
intros_11: fixes P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and sh :: "sheap"   and E :: "char list \<Rightarrow> ty option"   and e\<^sub>1 :: "char list exp"   and C :: "char list"   and F :: "char list"   and T :: "ty"   and D :: "char list"   and e\<^sub>2 :: "char list exp"   and T\<^sub>2 :: "ty" assumes "P,E,h,sh \<turnstile> e\<^sub>1 :' Class C"   and "P \<turnstile> C has F,NonStatic:T in D"   and "P,E,h,sh \<turnstile> e\<^sub>2 :' T\<^sub>2"   and "P \<turnstile> T\<^sub>2 \<le> T" shows "P,E,h,sh \<turnstile> e\<^sub>1\<bullet>F{D} := e\<^sub>2 :' Void"
zero_in_Field: fixes r :: "('c \<times> 'c) set" assumes "wo_rel r"   and "Field r \<noteq> {}" shows "zero r \<in> Field r"
bij_lens_iff_pbij_vwb: fixes X :: "'b \<Longrightarrow> 'c" shows "bij_lens X = (pbij_lens X \<and> vwb_lens X)"
vpairs_vinsert: fixes a :: "V"   and b :: "V"   and A :: "V" shows "vpairs (vinsert \<langle>a, b\<rangle> A) = set {\<langle>a, b\<rangle>} \<union>\<^sub>\<circ> vpairs A"
le_name_code_2: fixes x :: "char"   and xs :: "char list" shows "(Name [] \<le> Name (x # xs)) = True"
simps_2: fixes x :: "'c"   and xs :: "'c list"   and index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and n :: "nat"   and mi :: "'d"   and ma :: "'d" shows "enum (x # xs) index key n mi ma = (let i = index key x n mi ma; ns = enum xs index key n mi ma in ns[i := Suc (ns ! i)])"
distinct_1: fixes x21 :: "dag"   and x22 :: "ref"   and x23 :: "dag" shows "Tip \<noteq> Node x21 x22 x23"
fdia_star_induct_eq: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "d z +  |x\<rangle> y = d y" shows " |x\<^sup>\<star>\<rangle> z \<le> d y"
sum_card_b: fixes K :: "nat set"   and j0 :: "nat"   and j :: "nat" assumes "K \<subseteq> {j0<..}"   and "finite K"   and "j0 \<le> card K"   and "j \<le> j0" shows "(\<Sum>i<j. card (b (enum K i) (j0, i))) = enum (d j0) j - enum (d j0) 0"
unclocked_single_eq: shows "Big_Step_Unclocked_Single.evaluate = Big_Step_Unclocked.evaluate"
F_def: shows "F \<equiv> \<lambda>l (dl, a0, b0, M). let (d, Md) = grab (nxt M (enum N (Suc (2 * Suc l)))) (Suc l); (a, Ma) = grab Md (Min d); Gb = bf Ma (ediff (dl(l := d))); dl' = dl(l := d) in (dl', a, fst \<circ> Gb, snd (Gb (l, l - 1)))"
heterogeneous\<^sub>l\<^sub>s\<^sub>t_def: fixes t :: "('fun, 'var) Term.term"   and \<A> :: "('a strand_label \<times> ('fun, 'var) strand_step) list"   and Secrets :: "('fun, 'var) Term.term set" shows "heterogeneous\<^sub>l\<^sub>s\<^sub>t t \<A> Secrets \<equiv> \<exists>l1 l2. \<exists>s1\<in>subterms t. \<exists>s2\<in>subterms t. l1 \<noteq> l2 \<and> proj_specific l1 s1 \<A> Secrets \<and> proj_specific l2 s2 \<A> Secrets"
map_minus_plus_commute: fixes g :: "'c \<Rightarrow> 'd option"   and h :: "'c \<Rightarrow> 'd option"   and f :: "'c \<Rightarrow> 'd option" assumes "dom g \<inter> dom h = {}" shows "f -- g ++ h = f ++ h -- g"
totalI4: fixes f :: "recf" assumes "recfn 4 f"   and "\<And>w x y z. eval f [w, x, y, z] \<down>" shows "Partial_Recursive.total f"
dom_comp: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "VoV.seq g f" shows "VoV.dom (VoV.comp g f) = VoV.dom f"
simps_1: fixes g :: "nat \<Rightarrow> nat"   and h :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"   and x :: "nat" shows "PrimRecOp_last g h x 0 = g x"
eq1: shows "adjoint (2 \<cdot>\<^sub>m tensor_P proj_psi (1\<^sub>m K) - 1\<^sub>m d) = 2 \<cdot>\<^sub>m tensor_P proj_psi (1\<^sub>m K) - 1\<^sub>m d"
hp_eq: shows "\<forall>x\<in>set ps. g x = h x"
ladder_stepdown_diff_def: fixes L :: "(nat \<times> nat \<times> nat) list" shows "ladder_stepdown_diff L = Suc (ladder_n L 0)"
lderivs_Zero: fixes xs :: "'a list" shows "lderivs xs Zero = Zero"
obs_consistent_med01x: shows "obs_consistent R01s med01s s0 l1"
weakly_bisimilar_symp: shows "symp (\<approx>\<cdot>)"
simps: fixes I :: "'a set"   and v :: "nat \<Rightarrow> 'b"   and cs :: "('a \<times> 'b ns_constraint) set" shows "(I, v) \<Turnstile>\<^sub>i\<^sub>n\<^sub>s\<^sub>s  cs = v \<Turnstile>\<^sub>n\<^sub>s\<^sub>s  (snd ` (cs \<inter> I \<times> UNIV))"
ultra: fixes P :: "nat \<Rightarrow> bool" shows "eventually P \<U> \<or> (\<forall>\<^sub>F x in \<U>. \<not> P x)"
disj_conj_distrib2: fixes y :: "'b"   and z :: "'b"   and x :: "'b" shows "y AND z OR x = (y OR x) AND (z OR x)"
is_msg_defs_1: fixes \<xi> :: "'b state_scheme" shows "is_pkt \<xi> \<equiv> case msg \<xi> of Pkt num' sid' \<Rightarrow> {\<xi>\<lparr>num := num', sid := sid'\<rparr>} | Newpkt nat1 nat2 \<Rightarrow> {}"
diff_eq_eq: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "(a - b = c) = (a = c + b)"
cases: fixes r :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and s :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and a1 :: "'f"   and a2 :: "'h" assumes "(r OO s) a1 a2" obtains a :: "'f" and b :: "'g" and c :: "'h" where "a1 = a" and "a2 = c" and "r a b" and "s b c"
inj_imp_image_ring_is_field: fixes f :: "'a \<Rightarrow> 'c" assumes "inj_on f (carrier R)" shows "field (image_ring f R)"
eval_r_s: fixes x :: "nat" shows "s x = eval r_s [x]"
DynCastReds: fixes e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and C :: "char list" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>" shows "P,E \<turnstile> \<langle>Cast C e,s\<rangle> \<rightarrow>* \<langle>Cast C e',s'\<rangle>"
simCases: fixes Q :: "pi"   and P :: "pi"   and Rel :: "(pi \<times> pi) set" assumes "\<And>Q' a x. \<lbrakk>Q \<longmapsto> a<\<nu>x> \<prec> Q'; x \<sharp> P\<rbrakk> \<Longrightarrow> \<exists>P'. P \<Longrightarrow>a<\<nu>x> \<prec> P' \<and> (P', Q') \<in> Rel"   and "\<And>Q' \<alpha>. Q \<longmapsto> \<alpha> \<prec> Q' \<Longrightarrow> \<exists>P'. P \<Longrightarrow>\<^sup>^\<alpha> \<prec> P' \<and> (P', Q') \<in> Rel" shows "P \<leadsto><Rel> Q"
floor_exists: fixes x :: "'b" shows "\<exists>z. of_int z \<le> x \<and> x < of_int (z + 1)"
finite_cartesian_productD2: fixes A :: "'b set"   and B :: "'c set" assumes "finite (A \<times> B)"   and "A \<noteq> {}" shows "finite B"
assms: shows "0 < sin (Re z) \<or> sin (Re z) = 0 \<and> 0 \<le> Im z * cos (Re z)"
length_vec: fixes A :: "'b tensor" shows "length (vec A) = prod_list (dims A)"
of_int_poly_power: fixes p :: "int poly"   and n :: "nat" shows "of_int_poly (p ^ n) = of_int_poly p ^ n"
disc_transfer_1: fixes R1 :: "'f \<Rightarrow> 'i \<Rightarrow> bool"   and R2 :: "'g \<Rightarrow> 'j \<Rightarrow> bool" shows "rel_fun (Resumption.resumption.rel_resumption R1 R2) (=) Resumption.resumption.is_Done Resumption.resumption.is_Done"
assoc: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "lcm (lcm a b) c = lcm a (lcm b c)"
expectation_X_Y: fixes M :: "'a measure"   and I :: "'b set"   and A :: "'b \<Rightarrow> 'a set" assumes "prob_space_with_indicators M I A" shows "prob_space_with_indicators.\<mu> M I A = (\<Sum>i\<in>I. integral\<^sup>L M (prob_space_with_indicators.X A i))"
perfect_sec_P1_def: fixes m1 :: "bool \<times> bool"   and m2 :: "bool" shows "OT_12.perfect_sec_P1 m1 m2 \<equiv> R1 m1 m2 = funct_OT_12 m1 m2 \<bind> (\<lambda>(s1, s2). S1 m1 s1)"
fds_const_mult: fixes a :: "'a"   and b :: "'a" shows "fds_const (a * b) = fds_const a * fds_const b"
included_project: fixes getIJ :: "(nat set \<times> 'b option list set) set \<Rightarrow> (nat set \<times> 'b option list set) set \<Rightarrow> nat set \<Rightarrow> nat set \<times> nat set"   and V :: "nat set"   and Q :: "(nat set \<times> 'b option list set) set" assumes "getIJ getIJ" shows "included V (New_max.projectQuery V Q)"
image_mset_subset_mono: fixes M :: "'b multiset"   and N :: "'b multiset"   and f :: "'b \<Rightarrow> 'c" assumes "M \<subset># N" shows "image_mset f M \<subset># image_mset f N"
inline_parametric': fixes S :: "'k \<Rightarrow> 'p \<Rightarrow> bool"   and C :: "'l \<Rightarrow> 'q \<Rightarrow> bool"   and R :: "'m \<Rightarrow> 'r \<Rightarrow> bool"   and C' :: "'n \<Rightarrow> 's \<Rightarrow> bool"   and R' :: "'o \<Rightarrow> 't \<Rightarrow> bool"   and A :: "'u \<Rightarrow> 'v \<Rightarrow> bool" shows "((S ===> C ===> rel_gpv'' (rel_prod R S) C' R') ===> rel_gpv'' A C R ===> S ===> rel_gpv'' (rel_prod A S) C' R') inline inline"
exec_step_ind_Putfield_Static: fixes r :: "val"   and stk :: "val list"   and a :: "nat"   and D :: "char list"   and fs :: "Objects.fields"   and h :: "heap"   and P :: "jvm_prog"   and F :: "char list"   and t :: "ty"   and C :: "char list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap" assumes "r = hd (tl stk)"   and "a = the_Addr r"   and "(D, fs) = the (h a)"   and "r \<noteq> Null"   and "P \<turnstile> D has F,Static:t in C" shows "exec_step_ind (StepI (Putfield F C)) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (\<lfloor>addr_of_sys_xcpt IncompatibleClassChangeError\<rfloor>, h, (stk, loc, C\<^sub>0, M\<^sub>0, pc, ics) # frs, sh)"
canonical_sequence_def: fixes A :: "('b \<Rightarrow> bool) set"   and \<sigma> :: "nat \<Rightarrow> 'b" shows "canonical_sequence A \<sigma> \<equiv> \<forall>m n. \<sigma> m ~A~ \<sigma> n \<longrightarrow> \<sigma> m = \<sigma> n"
disjE1: fixes P :: "bool"   and Q :: "bool" assumes "P \<or> Q" obtains "P"   | "\<not> P" and "Q"
mgp_vertices_nonempty: fixes g :: "graph"   and f :: "face" assumes "minGraphProps g"   and "f \<in> \<F> g" shows "vertices f \<noteq> []"
pw_Sup_nofail: fixes X :: "'b nres set" shows "nofail (Sup X) = (\<forall>x\<in>X. nofail x)"
bitxor_one: fixes y :: "bit" shows "(\<one> \<oplus>\<^sub>b y) = (\<not>\<^sub>b y)"
strongly_accepts_def: fixes \<F> :: "nat set set"   and S :: "nat set"   and M :: "nat set" shows "strongly_accepts \<F> S M \<equiv> \<forall>N\<subseteq>M. rejects \<F> S N \<longrightarrow> finite N"
multithreaded_init_fin: shows "multithreaded if.init_fin_final if.init_fin"
Incremental_Atom_Ops_For_NS_Constraint_Ops_axioms: shows "Incremental_Atom_Ops_For_NS_Constraint_Ops init_s assert_s check_s solution_s checkpoint_s backtrack_s invariant_s checked_s preprocess"
simps_15: fixes e\<^sub>1 :: "char list exp"   and C :: "char list"   and V :: "char list"   and e\<^sub>2 :: "char list exp" shows "fv (try e\<^sub>1 catch(C V) e\<^sub>2) = fv e\<^sub>1 \<union> (fv e\<^sub>2 - {V})"
somewhere_leq: fixes v :: "view"   and v' :: "view" shows "(v \<le> v') = (\<exists>v1 v2 v3 vl vr vu vd. v'=vl\<parallel>v1 \<and> v1=v2\<parallel>vr \<and> v2=vd--v3 \<and> v3=v--vu)"
ot: fixes \<eta> :: "nat" shows "ot (\<G> \<eta>)"
relInvImage_def: fixes A :: "'c set"   and R :: "('d \<times> 'd) set"   and f :: "'c \<Rightarrow> 'd" shows "relInvImage A R f \<equiv> {(a1, a2) |a1 a2. a1 \<in> A \<and> a2 \<in> A \<and> (f a1, f a2) \<in> R}"
af_letter_subformulas\<^sub>\<nu>: fixes \<phi> :: "'b ltln"   and \<xi> :: "'b set" shows "subformulas\<^sub>\<nu> (af_letter \<phi> \<xi>) = subformulas\<^sub>\<nu> \<phi>"
parallel_interchange: fixes c\<^sub>0 :: "'a::refinement_lattice"   and c\<^sub>1 :: "'a::refinement_lattice"   and d\<^sub>0 :: "'a::refinement_lattice"   and d\<^sub>1 :: "'a::refinement_lattice" shows "((c\<^sub>0::'a::refinement_lattice) \<parallel> (c\<^sub>1::'a::refinement_lattice)) \<iinter> ((d\<^sub>0::'a::refinement_lattice) \<parallel> (d\<^sub>1::'a::refinement_lattice)) \<sqsubseteq> c\<^sub>0 \<iinter> d\<^sub>0 \<parallel> c\<^sub>1 \<iinter> d\<^sub>1"
inj_map: fixes f :: "'b \<Rightarrow> 'c" assumes "inj f" shows "inj (map f)"
function_mult_r_distr: fixes x :: "'c \<Rightarrow> 'a"   and y :: "'c \<Rightarrow> 'a"   and z :: "'c \<Rightarrow> 'a" assumes "x \<in> carrier F"   and "y \<in> carrier F"   and "z \<in> carrier F" shows "(x \<oplus>\<^bsub>F\<^esub> y) \<otimes>\<^bsub>F\<^esub> z = x \<otimes>\<^bsub>F\<^esub> z \<oplus>\<^bsub>F\<^esub> y \<otimes>\<^bsub>F\<^esub> z"
topology_inverse': fixes U :: "'c set \<Rightarrow> bool" assumes "istopology U" shows "openin (topology U) = U"
leq_trans: fixes leq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and lub :: "'f set \<Rightarrow> 'f"   and x :: "'f"   and y :: "'f"   and z :: "'f" assumes "partial_function_definitions leq lub"   and "leq x y"   and "leq y z" shows "leq x z"
assms_2: shows "\<langle>c \<Leftarrow> t\<rangle>\<^bsub>n\<^esub> \<le> n'"
fmrel_on_fsetD: fixes x :: "'d"   and S :: "'d fset"   and P :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and m :: "('d, 'e) fmap"   and n :: "('d, 'f) fmap" assumes "x |\<in>| S"   and "fmrel_on_fset S P m n" shows "rel_option P (fmlookup m x) (fmlookup n x)"
msubstltneg_nb: fixes t :: "tm"   and a :: "poly"   and r :: "tm"   and c :: "poly" assumes "tmbound0 t"   and "islin (Lt (CNP 0 a r))" shows "bound0 (msubstltneg c t a r)"
ad_rel_def: fixes R :: "('c \<times> 'd) set" shows "ad_rel R = {(x, x) |x. \<nexists>y. s2p R (x, y)}"
symI: fixes r :: "('c \<times> 'c) set" assumes "\<And>a b. (a, b) \<in> r \<Longrightarrow> (b, a) \<in> r" shows "sym r"
wlsAbs_castAbs: fixes xs :: "'varSort"   and s :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set" assumes "wlsAbs (xs, s) A" shows "wls s (snd (castAbs xs s A)) \<and> Abs xs (fst (castAbs xs s A)) (snd (castAbs xs s A)) = A"
distinct_concat_idx_disjoint: fixes xs :: "'a list list"   and i :: "nat"   and j :: "nat" assumes "distinct (concat xs)"   and "i < length xs"   and "j < length xs"   and "i < j" shows "set (xs ! i) \<inter> set (xs ! j) = {}"
hta_\<alpha>_def: fixes H :: "('a, 'b, 'c) hashedTa_scheme" shows "hta_\<alpha> H = \<lparr>ta_initial = hs.\<alpha> (hta_Qi H), ta_rules = ls.\<alpha> (hta_\<delta> H)\<rparr>"
less_aexp_def: fixes a1 :: "'b aexp"   and a2 :: "'b aexp" shows "(a1 < a2) = (let h1 = height a1; h2 = height a2 in if h1 = h2 then AExp_Lexorder.less_aexp_aux a1 a2 else h1 < h2)"
0: shows "0 \<le> (x - y)\<^sup>2"
h'_def: shows "h' = (\<lambda>x. \<Sum>t = 0..k. real (k choose t) * ln 2 ^ (k - t) * h t x)"
Gauss_Jordan_PA_eq: fixes A :: "(('b, 'c) vec, 'd) vec" shows "snd (Gauss_Jordan_PA A) = Gauss_Jordan A"
clock_repMC_S5n: fixes X :: "'s set" shows "S5n (clock_repMC X)"
lowner_lub_unique: fixes dim :: "nat"   and f :: "nat \<Rightarrow> complex mat" assumes "matrix_seq dim f" shows "\<exists>!M. matrix_seq.lowner_is_lub f M"
sorted_ConsI: fixes cmp :: "'b comparator"   and xs :: "'b list"   and x :: "'b" assumes "Sorting_Algorithms.sorted cmp xs"   and "\<And>y ys. xs = y # ys \<Longrightarrow> compare cmp x y \<noteq> Greater" shows "Sorting_Algorithms.sorted cmp (x # xs)"
simps: fixes X :: "'b \<Rightarrow> bool"   and I :: "'b \<Rightarrow> intv"   and J :: "'b \<Rightarrow> 'b \<Rightarrow> intv'"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b \<Rightarrow> real" shows "regionp X I J r a = (\<exists>u X\<^sub>0. a = u \<and> (\<forall>x\<in>{x. X x}. 0 \<le> u x) \<and> (\<forall>x\<in>{x. X x}. intv_elem x u (I x)) \<and> X\<^sub>0 = {x. X x \<and> (\<exists>d. I x = Intv d)} \<and> (\<forall>x\<in>X\<^sub>0. \<forall>y\<in>X\<^sub>0. r x y = (frac (u x) \<le> frac (u y))) \<and> (\<forall>x\<in>{x. X x}. \<forall>y\<in>{x. X x}. isGreater (I x) \<or> isGreater (I y) \<longrightarrow> intv'_elem x y u (J x y)))"
group_l_invI: assumes "\<And>x. x \<in> carrier (alt_group n) \<Longrightarrow> \<exists>y\<in>carrier (alt_group n). y \<otimes>\<^bsub>alt_group n\<^esub> x = \<one>\<^bsub>alt_group n\<^esub>" shows "Group.group (alt_group n)"
simps_6: fixes P :: "pol"   and Q :: "pol" shows "isnorm (PX P 0 Q) = False"
non_strict_dominate_def: fixes G :: "nat graph_rec"   and n1 :: "nat"   and n2 :: "nat" assumes "cfg_doms G" shows "cfg_doms.non_strict_dominate G n1 n2 \<equiv> \<exists>pa. path_entry (g_E G) pa n2 \<and> n1 \<notin> set pa"
rel_stateT_def: fixes S :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and M :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and m :: "('g, 'i) stateT"   and m' :: "('h, 'j) stateT" shows "rel_stateT S M m m' = (S ===> M) (run_state m) (run_state m')"
init_neq_zero: fixes f :: "nat \<Rightarrow> nat option"   and n :: "nat" shows "f \<triangleright> n \<noteq> 0"
extend_indets_ideal: fixes F :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set" shows "extend_indets ` ideal F = ideal (extend_indets ` F) \<inter> P[- {None}]"
induct: fixes x1 :: "any_unop + boolean_unop + numeric_unop + string_unop + collection_unop"   and x2 :: "call_kind"   and x3 :: "'a OCL_Types.type"   and x4 :: "'a OCL_Types.type"   and P :: "any_unop + boolean_unop + numeric_unop + string_unop + collection_unop \<Rightarrow> call_kind \<Rightarrow> 'a OCL_Types.type \<Rightarrow> 'a OCL_Types.type \<Rightarrow> bool" assumes "unop_type x1 x2 x3 x4"   and "\<And>op \<tau> \<sigma>. any_unop_type op \<tau> \<sigma> \<Longrightarrow> P (Inl op) DotCall \<tau> \<sigma>"   and "\<And>op \<tau> \<sigma>. boolean_unop_type op \<tau> \<sigma> \<Longrightarrow> P (Inr (Inl op)) DotCall \<tau> \<sigma>"   and "\<And>op \<tau> \<sigma>. numeric_unop_type op \<tau> \<sigma> \<Longrightarrow> P (Inr (Inr (Inl op))) DotCall \<tau> \<sigma>"   and "\<And>op \<tau> \<sigma>. string_unop_type op \<tau> \<sigma> \<Longrightarrow> P (Inr (Inr (Inr (Inl op)))) DotCall \<tau> \<sigma>"   and "\<And>op \<tau> \<sigma>. collection_unop_type op \<tau> \<sigma> \<Longrightarrow> P (Inr (Inr (Inr (Inr op)))) ArrowCall \<tau> \<sigma>" shows "P x1 x2 x3 x4"
REP_simps_2: fixes A :: "('a, 'a set) Maps.arr" assumes "Maps.arr A" shows "src (Maps.REP A) = Maps.Dom A"
rorc_channel_recv_def: fixes s :: "(bool \<times> unit) \<times> (bool list \<Rightarrow> bool list option) \<times> (bool list \<times> bool list) cstate"   and q :: "unit" shows "local.rorc_channel_recv s q \<equiv> \<dagger>\<dagger>channel.recv_oracle s () \<bind> (\<lambda>(m, s). case m of None \<Rightarrow> return_spmf (None, s) | Some (a, m) \<Rightarrow> \<dagger>(rorc.rnd_oracle (rnd \<eta>))\<dagger> s m \<bind> (\<lambda>(r, s). mac \<eta> r m \<bind> (\<lambda>a'. return_spmf (if a' = a then Some m else None, s))))"
converse_join: fixes r :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and s :: "'b \<Rightarrow> 'a \<Rightarrow> bool" shows "(r \<squnion> s)\<inverse>\<inverse> = r\<inverse>\<inverse> \<squnion> s\<inverse>\<inverse>"
rational_root_free_degree_iff: fixes p :: "int poly"   and x :: "'a" assumes "root_free (of_int_poly p)"   and "ipoly p x = (0::'a)" shows "(x \<in> \<rat>) = (degree p = 1)"
assms: shows "(p', i) # L = chop (to_list v p)"
cases: fixes x :: "('c \<Rightarrow> 'c \<Rightarrow> bool) \<times> 'c list" obtains P :: "'c \<Rightarrow> 'c \<Rightarrow> bool" where "x = (P__, [])"   | P :: "'c \<Rightarrow> 'c \<Rightarrow> bool" and xa :: "'c" and ys :: "'c list" where "x = (P, xa # ys)"
assms_2: shows "q = (1 + r - d) / (u - d)"
finite_abs_reach: fixes \<phi> :: "'b ltl" shows "finite (af_G_abs.abs_reach (Abs \<phi>))"
fair_derive_saturated_upto: fixes \<Gamma> :: "'b inference set"   and Rf :: "'b literal multiset set \<Rightarrow> 'b literal multiset set"   and Ri :: "'b literal multiset set \<Rightarrow> 'b inference set"   and Ns :: "'b literal multiset set llist" assumes "sat_preserving_effective_redundancy_criterion \<Gamma> Rf Ri"   and "chain (redundancy_criterion.derive \<Gamma> Rf) Ns"   and "effective_redundancy_criterion.fair_clss_seq \<Gamma> Rf Ri Ns" shows "redundancy_criterion.saturated_upto \<Gamma> Rf Ri (Liminf_llist Ns)"
incomplete_alt_imp: fixes bl :: "'a set" assumes "int (card bl) < \<v>"   and "bl \<in># \<B>" shows "incomplete_block bl"
assms: shows "\<Psi> : Q \<rhd> P \<Longrightarrow>\<alpha> \<prec> P'"
make_affine_def: fixes R :: "('c, 'd) ring_scheme"   and p :: "'c \<times> 'c \<times> 'c" assumes "field R" shows "field.make_affine R p = (let (x, y, z) = p in if z = \<zero>\<^bsub>R\<^esub> then Infinity else Point (x \<oslash>\<^bsub>R\<^esub> z) (y \<oslash>\<^bsub>R\<^esub> z))"
bv_sadd_def: fixes w1 :: "bit list"   and w2 :: "bit list" shows "bv_sadd w1 w2 = int_to_bv (bv_to_int w1 + bv_to_int w2)"
derived_is_wellformed: fixes b :: "'a set" assumes "b \<in># \<B>\<^sup>D" shows "b \<subseteq> bl"
finite_supp: fixes permute :: "perm \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.fs permute" shows "finite (pt.supp permute x)"
functor_axioms: shows "functor (\<cdot>\<^sub>A\<^sub>1\<^sub>x\<^sub>A\<^sub>2) (\<cdot>\<^sub>B) G"
x: shows "x \<in> {- exp (- 1)<..<0}"
domains_comp: fixes g :: "'a"   and f :: "'a" assumes "B.seq g f" shows "B.domains (g \<cdot>\<^sub>B f) = B.domains f"
continuous_on_norm_id: fixes S :: "'c set" shows "continuous_on S norm"
assms: fixes Y :: "nat \<Rightarrow> 'a" assumes "chain Y" shows "f (\<Squnion>i. Y i) = \<Union> (f ` range Y)"
ale_diff_pos: fixes x :: "ant"   and y :: "ant" assumes "x \<le> y" shows "0 \<le> y - x"
linear: fixes f :: "'a \<Rightarrow> 'b" assumes "bounded_linear f" shows "linear f"