hcju/isabelleps huggingface dataset:
of_nat_0: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d" assumes "class.semiring_1 one times plus zero" shows "semiring_1.of_nat one plus zero 0 = zero"
diff_single_trivial: fixes x :: "'c"   and M :: "'c multiset" assumes "x \<notin># M" shows "M - {#x#} = M"
linear_dependent_inj_imageD: fixes f :: "'d \<Rightarrow> 'e"   and s :: "'d set" assumes "clinear f"   and "cdependent (f ` s)"   and "inj_on f (cspan s)" shows "cdependent s"
some_GB_isGB_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat" assumes "dickson_grading d"   and "finite (component_of_term ` Keys F)"   and "F \<subseteq> dgrad_p_set d m" shows "is_Groebner_basis (some_GB F)"
Derives1_of_singleton: fixes N :: "symbol"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and \<alpha> :: "symbol list" assumes "Derives1 [N] i r \<alpha>" shows "i = 0 \<and> r = (N, \<alpha>)"
assms_3: shows "snd (rep_nat_term u) < snd (rep_nat_term v)"
Transset_def: fixes x :: "hf" shows "Transset x \<equiv> \<forall>y. y \<^bold>\<in> x \<longrightarrow> y \<le> x"
h_def: shows "h \<equiv> \<lambda>x. - g x - f x + f x * ln (b * x) powr (- e / 2) + f x * g x"
Acyclic_Graph_axioms: shows "Acyclic_Graph nodes inPorts outPorts vertices nodeOf edges hyps"
AxSig: fixes Ax :: "('a, 'b) Axioms" assumes "Axioms Ax" shows "Signature aS\<^bsub>Ax\<^esub>"
update_convs_1: fixes p_iiface' :: "char list \<Rightarrow> char list"   and p_iiface :: "char list"   and p_oiface :: "char list"   and p_src :: "'a word"   and p_dst :: "'a word"   and p_proto :: "8 word"   and p_sport :: "16 word"   and p_dport :: "16 word"   and p_tcp_flags :: "tcp_flag set"   and p_payload :: "char list"   and more :: "'b" shows "p_iiface_update (p_iiface'::char list \<Rightarrow> char list) \<lparr>p_iiface = p_iiface::char list, p_oiface = p_oiface::char list, p_src = p_src::'a word, p_dst = p_dst::'a word, p_proto = p_proto::8 word, p_sport = p_sport::16 word, p_dport = p_dport::16 word, p_tcp_flags = p_tcp_flags::tcp_flag set, p_payload = p_payload::char list, \<dots> = more::'b\<rparr> = \<lparr>p_iiface = p_iiface' p_iiface, p_oiface = p_oiface, p_src = p_src, p_dst = p_dst, p_proto = p_proto, p_sport = p_sport, p_dport = p_dport, p_tcp_flags = p_tcp_flags, p_payload = p_payload, \<dots> = more\<rparr>"
simps_3: fixes i :: "nat" shows "Exit \<oplus> i = Exit"
elims: fixes x :: "graph"   and y :: "nat" assumes "Dijkstra.size x = y" obtains G :: "nat list list" where "x = Graph G" and "y = length G"
assms_2: shows "\<exists>cnv. \<forall>\<D>. (\<forall>\<eta>. \<I>_ideal \<eta> \<oplus>\<^sub>\<I> \<I>_common \<eta> \<turnstile>g \<D> \<eta> \<surd>) \<longrightarrow> (\<forall>\<eta>. interaction_any_bounded_by (\<D> \<eta>) (bound \<eta>)) \<longrightarrow> (\<forall>\<eta>. lossless \<longrightarrow> plossless_gpv (\<I>_ideal \<eta> \<oplus>\<^sub>\<I> \<I>_common \<eta>) (\<D> \<eta>)) \<longrightarrow> (\<forall>\<eta>. wiring (\<I>_ideal \<eta>) (\<I>_real \<eta>) (cnv \<eta>) (w \<eta>)) \<and> Negligible.negligible (\<lambda>\<eta>. advantage (\<D> \<eta>) (ideal_resource \<eta>) (cnv \<eta> |\<^sub>= 1\<^sub>C \<rhd> real_resource \<eta>))"
leg1_commutes: shows "cod.leg1 \<cdot> chine = dom.leg1"
encode_all_frame_axioms_def: fixes \<Pi> :: "'b strips_problem"   and t :: "nat" shows "encode_all_frame_axioms \<Pi> t \<equiv> let l = List.product [0..<t] (\<Pi>\<^sub>\<V>) in \<^bold>\<And>map (\<lambda>(k, v). encode_negative_transition_frame_axiom \<Pi> k v) l @ map (\<lambda>(k, v). encode_positive_transition_frame_axiom \<Pi> k v) l"
simps_64: fixes f :: "'b \<Rightarrow> 'a"   and x3 :: "'b" shows "map\<^sub>f\<^sub>m f (Atom x3) = Atom (f x3)"
DiaP'': fixes a :: "'c"   and ps :: "('b, 'c) fm list"   and branch :: "(('b, 'c) fm list \<times> 'c) list"   and p :: "('b, 'c) fm"   and i :: "'c"   and A :: "'c set" assumes "(\<^bold>\<diamond> p) at a in' set ((ps, a) # branch)"   and "i \<notin> A \<union> branch_nominals ((ps, a) # branch)"   and "\<nexists>a. p = Nom a"   and "finite A"   and "A \<turnstile> ((\<^bold>@ i p) # (\<^bold>\<diamond> Nom i) # ps, a) # branch" shows "A \<turnstile> (ps, a) # branch"
b_e_type_value_list_2: fixes \<C> :: "t_context"   and es :: "b_e list"   and v :: "v"   and ts :: "t list"   and ts' :: "t list"   and t :: "t" assumes "\<C> \<turnstile> es @ [C v] : ts _> ts' @ [t]" shows "typeof v = t"
subst_cls_mset_add_mset: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and C :: "'b literal multiset"   and CC :: "'b literal multiset multiset"   and \<sigma> :: "'c" shows "substitution_ops.subst_cls_mset subst_atm (add_mset C CC) \<sigma> = add_mset (substitution_ops.subst_cls subst_atm C \<sigma>) (substitution_ops.subst_cls_mset subst_atm CC \<sigma>)"
pcompose_def: fixes p :: "'b poly"   and q :: "'b poly" shows "p \<circ>\<^sub>p q = fold_coeffs (\<lambda>a c. [:a:] + q * c) p 0"
A: fixes zs :: "'a list" shows "(\<forall>u\<in>sinks_aux_less I D U ys. \<forall>z\<in>Z \<union> set zs. (u, D z) \<notin> I) \<longrightarrow> (xs @ ipurge_tr_aux I D U ys @ zs, Z) \<in> failures P \<longrightarrow> xs @ ys \<in> traces P \<longrightarrow> (xs @ ys @ zs, Z) \<in> failures P"
hyps: shows "(\<Sum>i<2 * n. if even i then f i else g i) = (\<Sum>i<n. f (2 * i)) + (\<Sum>i<n. g (2 * i + 1))"
stutter_extend_simps_sa: fixes G :: "('a, 'b, 'c) sa_rec_scheme" shows "sa_L (stutter_extend G) = sa_L G"
total_order_total: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier L"   and "y \<in> carrier L" shows "x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
pth_c_4: fixes c :: "real"   and x :: "'c"   and w :: "'c"   and d :: "real"   and y :: "'c"   and z :: "'c" shows "c *\<^sub>R x + w + (d *\<^sub>R y + z) \<equiv> d *\<^sub>R y + (c *\<^sub>R x + w + z)"
prems: fixes i :: "'a" assumes "i \<in> insert a S" shows "finite (T i)"
0: fixes x :: "nat" assumes "x \<le> p" shows "simplex_cone (p - Suc 0) v (singular_face p x (oriented_simplex p l)) = oriented_simplex p (\<lambda>j. if j < Suc x then if j = 0 then v else l (j - 1) else if Suc j = 0 then v else l (Suc j - 1))"
set_image_mset: fixes f :: "'b \<Rightarrow> 'a"   and M :: "'b multiset" shows "set_mset (image_mset f M) = f ` set_mset M"
cases: fixes x :: "'b heap" obtains uu_ :: "'b" and hs :: "'b heap list" where "x = Hp uu_ hs"   | "x = heap.Empty"
nn_integral_PiM_Suc: fixes N :: "nat \<Rightarrow> 'b measure"   and f :: "(nat \<Rightarrow> 'b) \<Rightarrow> ennreal"   and M :: "'b measure" assumes "\<And>i. sigma_finite_measure (N i)"   and "f \<in> borel_measurable (Pi\<^sub>M V N)" shows "integral\<^sup>N (distr (Pi\<^sub>M (Suc ` V) (case_nat M N)) (Pi\<^sub>M V N) (\<lambda>\<sigma>. \<sigma> \<circ> Suc)) f = integral\<^sup>N (Pi\<^sub>M V N) f"
pair_less_def: shows "pair_less = less_than <*lex*> less_than"
set_list_union: fixes xs :: "'a list"   and ys :: "'a list" shows "set (list_union xs ys) = set xs \<union> set ys"
lift_is_hom: fixes f :: "'d \<Rightarrow> 'a"   and gens :: "'d set" assumes "f \<in> gens \<rightarrow> carrier G" shows "lift f \<in> hom \<F>\<^bsub>gens\<^esub> G"
consistent_imp_is_solution': fixes A :: "(('b, 'c) vec, 'd) vec"   and b :: "('b, 'd) vec" assumes "consistent A b" shows "is_solution (solve_consistent_rref (fst (solve_system A b)) (snd (solve_system A b))) A b"
image_Suc_atMost: fixes n :: "nat" shows "Suc ` {..n} = {1..Suc n}"
is_singleton_def: fixes A :: "'b set" shows "is_singleton A = (\<exists>x. A = {x})"
IR1_1: shows "callee_invariant_on (callee_of_rest rest1) IR1 (\<I>RA \<oplus>\<^sub>\<I> \<I>RU)"
has_field_derivative_sin_z_over_z: fixes A :: "'a set" shows "((\<lambda>z. if z = (0::'a) then 1::'a else sin z / z) has_field_derivative (0::'a)) (at (0::'a) within A)"
bij_is_surj: fixes f :: "'d \<Rightarrow> 'e" assumes "bij f" shows "surj f"
Ri_effective: fixes \<gamma> :: "'a inference"   and N :: "'a literal multiset set" assumes "\<gamma> \<in> \<Gamma>"   and "concl_of \<gamma> \<in> N \<union> Rf N" shows "\<gamma> \<in> Ri N"
hyps_2: shows "fcompose (p1 * p2) q1 q2 = fcompose p1 q1 q2 * fcompose p2 q1 q2"
extend_list_of_filtered_set: fixes S :: "'c set"   and xs' :: "'c list"   and P :: "'c \<Rightarrow> bool" assumes "finite S"   and "distinct xs'"   and "set xs' = {x \<in> S. P x}" obtains xs :: "'c list" where "xs' = filter P xs" and "distinct xs" and "set xs = S"
cgenideal_eq_genideal: fixes R :: "('a, 'b) ring_scheme"   and i :: "'a" assumes "cring R"   and "i \<in> carrier R" shows "PIdl\<^bsub>R\<^esub> i = Idl\<^bsub>R\<^esub> {i}"
least_method_is_type: fixes 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 C :: "char list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and m :: "char list list \<times> expr"   and Cs :: "char list list"   and wf_md :: "(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 \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool" assumes "P \<turnstile> C has least M = (Ts, T, m) via Cs"   and "wf_prog wf_md P" shows "is_type P T"
star_circ_simulate_right_plus: fixes z :: "'a"   and x :: "'a"   and y :: "'a"   and w :: "'a" assumes "z * x \<le> y\<^sup>+ * z \<squnion> w" shows "z * x\<^sup>\<star> \<le> y\<^sup>\<star> * (z \<squnion> w * x\<^sup>\<star>)"
r': fixes n :: "real" shows "sum_upto (\<lambda>x. ln (real x)) n = n * ln n + r' n"
allow_pfun_def: fixes pf :: "'c \<Rightarrow> 'd option" shows "A\<^sub>p pf \<equiv> \<lambda>x. case pf x of \<bottom> \<Rightarrow> \<bottom> | \<lfloor>y\<rfloor> \<Rightarrow> \<lfloor>allow y\<rfloor>"
intros_2: fixes a :: "letter"   and ws :: "letter list list"   and zs :: "letter list list"   and w :: "letter list" assumes "T a ws zs" shows "T a (w # ws) ((a # w) # zs)"
assms_1: shows "!P \<longmapsto>\<alpha> \<prec> P'"
cut_less_Int_conv: fixes I :: "'b set"   and t :: "'b" shows "I \<down>< t = I \<inter> {..<t}"
iwlsFSwSb_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'a, 'b, 'c, 'd) model_scheme" shows "iwlsFSwSb MOD \<equiv> iwlsFSw MOD \<and> igSubstAllIPresIGWlsAll MOD \<and> igSubstCls MOD"
derangements_in_image: fixes d :: "nat \<Rightarrow> nat"   and A :: "nat set"   and x :: "nat" assumes "d \<in> derangements A"   and "x \<in> A" shows "d x \<in> A"
zorder_inverse: fixes f :: "complex \<Rightarrow> complex"   and z :: "complex" assumes "isolated_singularity_at f z"   and "not_essential f z"   and "\<exists>\<^sub>F w in at z. f w \<noteq> 0" shows "zorder (\<lambda>w. inverse (f w)) z = - zorder f z"
localSS'_aux_b: shows "(\<lambda>(x, u, v) ab. u = (0::'b) \<and> (case ab of (y, u', v') \<Rightarrow> u' = Suc 0 \<and> v = v')) OO (\<lambda>(x, u, v) (y, u', v'). u = u') = (\<lambda>(x, u, v) (y, u', v'). u = (0::'b) \<and> u' = 1)"
unsat: shows "I \<subseteq> fst ` set cs \<and> (\<nexists>v. (I, v) \<Turnstile>\<^sub>i\<^sub>n\<^sub>s\<^sub>s  set (to_ns cs)) \<and> (distinct_indices_ns (set (to_ns cs)) \<longrightarrow> (\<forall>J\<subset>I. \<exists>v. (J, v) \<Turnstile>\<^sub>i\<^sub>n\<^sub>s\<^sub>s  set (to_ns cs)))"
maxim_equality: fixes A :: "'a set"   and x :: "'a" assumes "isMaxim A x" shows "maxim A = x"
B_out: fixes x :: "'v"   and y :: "'v" assumes "x \<in> B \<Gamma>" shows "\<not> edge \<Gamma> x y"
f_def: shows "f \<equiv> (\<lambda>x. if x \<in> m ` C then Inl (the_inv_into C m x) else Inr x) \<circ> h"
eq: shows "Pi\<^sub>E I F = Pi\<^sub>E I F'"
interval_integral_eq_integral: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a" assumes "a \<le> b"   and "set_integrable lborel {a..b} f" shows "interval_lebesgue_integral lborel (ereal a) (ereal b) f = integral {a..b} f"
simps_11: fixes b :: "bool"   and c :: "expr"   and e1 :: "expr"   and e2 :: "expr" shows "assigns_if b (c ? e1 : e2) = assignsE c \<union> (case constVal c of None \<Rightarrow> assigns_if b e1 \<inter> assigns_if b e2 | Some bv \<Rightarrow> case the_Bool bv of True \<Rightarrow> assigns_if b e1 | False \<Rightarrow> assigns_if b e2)"
assms_1: shows "(x, y) \<in> FL_Validity.alpha_Tree_rel"
RPDs_code: fixes r :: "mregex" shows "RPDs r = saturate (\<lambda>S. insert r S \<union> (insert r S \<bind> RPD)) {}"
total_on_subset: fixes B :: "'a set"   and R :: "('a \<times> 'a) set"   and A :: "'a set" assumes "total_on B R"   and "A \<subseteq> B" shows "total_on A R"
normNum_isnormNum: fixes x :: "int \<times> int" shows "isnormNum (normNum x)"
cases: fixes a :: "'transition" assumes "a \<in> executable" obtains p :: "'state" and a :: "'transition" where "aa__ = a" and "p \<in> nodes" and "enabled a p"
assms_1: fixes p :: "'a" assumes "p \<in># P" shows "prime p"
is_basis_alt: fixes V :: "'b set"   and B :: "'b set" assumes "local.subspace V" shows "is_basis V B = (local.independent B \<and> local.span B = V)"
trans: fixes b :: "'c"   and a :: "'c"   and c :: "'c" assumes "b \<sqsubseteq> a"   and "c \<sqsubseteq> b" shows "c \<sqsubseteq> a"
blinding_of_list_hash: fixes bo :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and h :: "'c \<Rightarrow> 'd" assumes "bo \<le> vimage2p h h (=)" shows "list_all2 bo \<le> vimage2p (map h) (map h) (=)"
pseudo_proj_False_preimage: fixes w :: "bool stream"   and n :: "nat" assumes "w = pseudo_proj_False n w" shows "pseudo_proj_False n -` {w} = (\<Inter>i\<in>{m. Suc m \<le> n}. (\<lambda>w. w !! i) -` {w !! i})"
take_bit_of_0: fixes n :: "nat" shows "take_bit n (0::'a) = (0::'a)"
valid_app_freshs_1: fixes \<Gamma> :: "(name \<times> ty) list"   and \<Delta> :: "(name \<times> ty) list" assumes "valid (\<Gamma> @ \<Delta>)" shows "supp \<Gamma> \<sharp>* \<Delta>"
subst_simp_fst: fixes P :: "'a trm"   and x :: "'a"   and M :: "'a trm" shows "Fst P[x ::= M] = Fst P[x ::= M]"
tendsto_add_const_iff: fixes c :: "'b"   and f :: "'a \<Rightarrow> 'b"   and d :: "'b"   and F :: "'a filter" shows "((\<lambda>x. c + f x) \<longlongrightarrow> c + d) F = (f \<longlongrightarrow> d) F"
intros: fixes g :: "'g"   and ns :: "'node list" assumes "g \<turnstile> Entry g-ns\<rightarrow>last ns" shows "g \<turnstile> ns\<Down>fold (step g) ns Map.empty"
PLemmas_9: fixes m :: "'a \<Rightarrow> 'b option" shows "dom m = {a. m a \<noteq> \<bottom>}"
minimalFace_def: shows "minimalFace \<equiv> minimal (length \<circ> vertices)"
2: shows "lincomb b A = lincomb b (S \<inter> A)"
Ch_XH_largest_Field: fixes ds :: "'b set"   and X :: "'a set"   and x :: "'a"   and h :: "'c" assumes "stable_on ds X"   and "x \<in> Ch h (XH_largest ds X)" shows "x \<in> Field (Pd (Xd x))"
unit_factor_Lcm: fixes A :: "'a set" shows "unit_factor (Lcm A) = (if Lcm A = (0::'a) then 0::'a else (1::'a))"
interleave_induct: fixes x3 :: "'b list"   and x1 :: "'b list"   and x2 :: "'b list"   and P :: "'b list \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "x3 \<in> x1 \<otimes> x2"   and "P [] [] []"   and "\<And>xs ys zs x. \<lbrakk>zs \<in> xs \<otimes> ys; P xs ys zs\<rbrakk> \<Longrightarrow> P (x # xs) ys (x # zs)"   and "\<And>xs ys zs x. \<lbrakk>zs \<in> xs \<otimes> ys; P xs ys zs\<rbrakk> \<Longrightarrow> P xs (x # ys) (x # zs)" shows "P x1 x2 x3"
connected_def: fixes v :: "'a"   and w :: "'a" shows "v \<rightarrow>\<^sup>* w \<equiv> \<exists>xs. walk xs \<and> xs \<noteq> [] \<and> hd xs = v \<and> last xs = w"
is_extensional: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and G :: "'a \<Rightarrow> 'b"   and \<tau> :: "'a \<Rightarrow> 'b"   and f :: "'a" assumes "natural_transformation A B F G \<tau>"   and "\<not> partial_magma.arr A f" shows "\<tau> f = partial_magma.null B"
epsclo_nextl: fixes Q :: "hf set"   and xs :: "'a list" shows "epsclo (nextl Q xs) = nextl Q xs"
refl_reflcl: fixes r :: "('b \<times> 'b) set" shows "refl (r\<^sup>=)"
SD_efficient_def: fixes p :: "'b pmf" shows "SD_efficient R p = (\<not> (\<exists>q\<in>lotteries_on alts. p \<prec>[Pareto (SD \<circ> R)] q))"
simps_8: fixes f :: "'f \<Rightarrow> 'e" shows "map_option f None = None"
map_update: fixes m :: "'m"   and a :: "'a"   and b :: "'b" assumes "invar m" shows "lookup (update a b m) = lookup m(a \<mapsto> b)"
max_ge_y: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and y :: "'c"   and x :: "'c" assumes "class.non_strict_order less_eq" shows "less_eq y (ord.max less_eq x y)"
\<tau>move2_AAssD1: fixes P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and a :: "'addr expr1"   and i :: "'addr expr1"   and e :: "'addr expr1"   and pc :: "nat"   and xcp :: "'addr option" assumes "\<tau>move2 P h stk (a\<lfloor>i\<rceil> := e) pc xcp"   and "pc < length (compE2 a)" shows "\<tau>move2 P h stk a pc xcp"
infinitesimal_quad_helper: fixes At :: "atomUni"   and p :: "real \<times> real \<times> real"   and D :: "real"   and C :: "real"   and A :: "real"   and B :: "real"   and x :: "real" assumes "At = LessUni p \<or> At = EqUni p"   and "D \<noteq> 0"   and "0 \<le> C" shows "(\<exists>y'>(A + B * sqrt C) / D. \<forall>x\<in>{(A + B * sqrt C) / D<..y'}. aEvalUni At x) = evalUni (substInfinitesimalQuadraticUni A B C D At) x"
gmnl_gmd: fixes d :: "('a, 'b) Digit" assumes "FingerTreeStruc.is_measured_digit d" shows "FingerTreeStruc.gmnl (FingerTreeStruc.digitToNlist d) = FingerTreeStruc.gmd d"
coeff_Poly_eq: fixes xs :: "'a list" shows "coeff (Poly xs) = nth_default (0::'a) xs"
cont_case_prod_applyD: fixes luba :: "('b \<times> 'c) set \<Rightarrow> 'b \<times> 'c"   and orda :: "'b \<times> 'c \<Rightarrow> 'b \<times> 'c \<Rightarrow> bool"   and lubb :: "'d set \<Rightarrow> 'd"   and ordb :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'e \<Rightarrow> 'd"   and y :: "'e" assumes "cont luba orda lubb ordb (\<lambda>x. (case x of (x, xa) \<Rightarrow> f x xa) y)" shows "cont luba orda lubb ordb (\<lambda>(a, b). f a b y)"
cut_same_stop: fixes x :: "'c"   and y :: "'c"   and xs :: "'c list" assumes "x \<noteq> y" shows "cut_same x (xs @- y ## sconst x) = xs @ [y]"
rel: shows "P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>RT<TRel> Q"
bind_def: fixes A :: "'e set"   and f :: "'e \<Rightarrow> 'd set" shows "A \<bind> f = {x. \<exists>B\<in>f ` A. x \<in> B}"
totalSubset: fixes A :: "nat set"   and B :: "nat set"   and valuation :: "Literal list" assumes "A \<subseteq> B"   and "CNF.total valuation B" shows "CNF.total valuation A"
path_Exit_source_2: fixes as :: "'edge list"   and n' :: "'node" assumes "(_Exit_) -as\<rightarrow>* n'" shows "as = []"
min_Least: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "class.wellorder less_eq less"   and "\<exists>n. P n"   and "\<exists>n. Q n" shows "ord.min less_eq (ord.Least less_eq P) (ord.Least less_eq Q) = ord.Least less_eq (\<lambda>n. P n \<or> Q n)"
inverse_matrix_def: fixes A :: "(('b, 'd) vec, 'c) vec" shows "inverse_matrix A = (if invertible A then Some (matrix_inv A) else None)"
sem_nonintDfmla: fixes env :: "pvar \<Rightarrow> nat" shows "sem nonintDfmla env = nonintSfmla"
pareto_losersE: fixes x :: "'alt" assumes "x \<in> pareto_losers R" obtains y :: "'alt" where "y \<in> alts" and "x \<prec>[Pareto R] y"
L_ss: fixes G :: "('a, 'b, 'c) gba_rec_scheme"   and q :: "'a"   and l :: "'b" assumes "gba G"   and "gba_L G q l" shows "q \<in> g_V G"
rep_eq: fixes x :: "'b blinop" shows "blinop_apply x = blinfun_apply (Rep_blinop x)"
distinct_smult_nonneg: fixes c :: "'b"   and lC :: "'b vec list" assumes "c \<noteq> (0::'b)"   and "distinct lC" shows "distinct (map ((\<cdot>\<^sub>v) c) lC)"
removel_correct_1: fixes s :: "'s" assumes "invar s"   and "\<alpha> s \<noteq> []" shows "fst (removel s) = hd (\<alpha> s)"
infsetsum_altdef': fixes A :: "'d set"   and B :: "'d set"   and f :: "'d \<Rightarrow> 'e" assumes "A \<subseteq> B" shows "infsetsum f A = set_lebesgue_integral (count_space B) A f"
i_aggregate_i_append_mult: fixes xs :: "'b list"   and m :: "nat"   and k :: "nat"   and f :: "nat \<Rightarrow> 'b"   and ag :: "'b list \<Rightarrow> 'b" assumes "length xs = m * k" shows "i_aggregate (xs \<frown> f) k ag = f_aggregate xs k ag \<frown> i_aggregate f k ag"
that: fixes a :: "V"   and b :: "V" assumes "x = [a, b]\<^sub>\<circ>"   and "a \<in>\<^sub>\<circ> A"   and "b \<in>\<^sub>\<circ> B" shows "thesis"
noStuck_Call_body: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and p :: "'e"   and bdy :: "('d, 'e, 'f) com"   and s :: "'d" assumes "\<Gamma> p = Some bdy" shows "\<Gamma>\<turnstile> \<langle>Call p,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck} = \<Gamma>\<turnstile> \<langle>the (\<Gamma> p),Normal s\<rangle> \<Rightarrow>\<notin>{Stuck}"
wbt_split_min: fixes t :: "('b \<times> nat) tree" assumes "t \<noteq> \<langle>\<rangle>"   and "wbt t" shows "wbt (snd (split_min t))"
1: shows "w \<sqinter> top * e * w\<^sup>T\<^sup>\<star> \<sqinter> - v\<^sup>T \<le> - v"
of_real_closed_segment: fixes x :: "real"   and a :: "real"   and b :: "real" shows "(of_real x \<in> closed_segment (of_real a) (of_real b)) = (x \<in> closed_segment a b)"
finite_convex_hull_iff_convex_hull_list: fixes Vs :: "'b vec set"   and n :: "nat"   and Vsl' :: "'b vec list" assumes "Vs \<subseteq> carrier_vec n"   and "Vs = set Vsl'" shows "gram_schmidt.convex_hull n Vs = gram_schmidt.convex_hull_list n Vsl'"
assms_1: shows "n'' -{V}as\<rightarrow>\<^sub>d\<^sub>d n'"
get_ancestors_si_def: shows "get_ancestors_si \<equiv> l_get_root_node_si\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_ancestors_si Shadow_DOM.get_parent get_host"
u_def: shows "u = Tensor.mat_of_cols_list 4 [[v $$ (0, 0) * w $$ (0, 0), v $$ (0, 0) * w $$ (1, 0), v $$ (1, 0) * w $$ (0, 0), v $$ (1, 0) * w $$ (1, 0)]]"
simps: fixes a1 :: "fterm literal set set"   and a2 :: "fterm literal set set" shows "resolution_step a1 a2 = ((\<exists>C\<^sub>1 Cs C\<^sub>2 L\<^sub>1 L\<^sub>2 \<sigma>. a1 = Cs \<and> a2 = Cs \<union> {resolution C\<^sub>1 C\<^sub>2 L\<^sub>1 L\<^sub>2 \<sigma>} \<and> C\<^sub>1 \<in> Cs \<and> C\<^sub>2 \<in> Cs \<and> applicable C\<^sub>1 C\<^sub>2 L\<^sub>1 L\<^sub>2 \<sigma>) \<or> (\<exists>C Cs C'. a1 = Cs \<and> a2 = Cs \<union> {C'} \<and> C \<in> Cs \<and> var_renaming_of C C'))"
self_in_upset: fixes x :: "'b" shows "x \<in> \<up>x"
einterval_Icc_approximation: fixes a :: "ereal"   and b :: "ereal" assumes "a < b" obtains u :: "nat \<Rightarrow> real" and l :: "nat \<Rightarrow> real" where "einterval a b = (\<Union>i. {l i..u i})" and "incseq u" and "decseq l" and "\<And>i. l i < u i" and "\<And>i. a < ereal (l i)" and "\<And>i. ereal (u i) < b" and "(\<lambda>x. ereal (l x)) \<longlonglongrightarrow> a" and "(\<lambda>x. ereal (u x)) \<longlonglongrightarrow> b"
offs_add: fixes i :: "nat"   and ns :: "nat list"   and k :: "nat" assumes "i < length ns" shows "offs ns k ! i = foldl (+) k (take i ns)"
\<sigma>_def: fixes f :: "'a \<times> 'a" shows "\<sigma> f \<equiv> if CC.arr f then \
[cod (fst f), cod (snd f)] \<cdot> (fst f \<otimes> snd f) else null"
Collect_empty_eq: fixes P :: "'g \<Rightarrow> bool" shows "(Collect P = {}) = (\<forall>x. \<not> P x)"
path_from_to_last': fixes v :: "'a"   and xs :: "'a list"   and x :: "'a"   and xs' :: "'a list"   and w :: "'a" assumes "v \<leadsto>(xs @ x # xs')\<leadsto>\<^bsub>H_x\<^esub> w" shows "w \<notin> set xs"
contractible_def: fixes S :: "'b set" shows "contractible S \<equiv> \<exists>a. homotopic_with_canon (\<lambda>x. True) S S id (\<lambda>x. a)"
simps_4: fixes uw :: "'BExp"   and p :: "('Var, 'AExp, 'BExp) Stmt" shows "has_annotated_stop (While uw p) = has_annotated_stop p"
mset_factors_imp_not_is_unit: fixes F :: "'a multiset"   and p :: "'a" assumes "mset_factors F p" shows "\<not> p dvd (1::'a)"
subset_gen: fixes J :: "'a set"   and P :: "'a set set"   and I :: "'a set" assumes "J \<in> P"   and "gen P I \<inter> J \<noteq> {}" shows "J \<subseteq> gen P I"
Lxy_set_filter: fixes xs :: "'c list"   and S :: "'c set" shows "set (Lxy xs S) = S \<inter> set xs"