hcju/isabelleps huggingface dataset:
nnfisp_nnfis_eq: fixes f :: "'b \<Rightarrow> real"   and M :: "'b set set \<times> ('b set \<Rightarrow> real)" shows "nnfisp f M = (\<lambda>x. x \<in> nnfis f M)"
geodesic_segment_dist'_2: fixes x :: "'a"   and z :: "'a"   and y :: "'a" assumes "dist x z = dist x y + dist y z" shows "{x--z} = {x--y} \<union> {y--z}"
alw_invs: fixes s :: "('b, 't, 'loc) Combined.configuration stream" assumes "FullSpec s" shows "alw (holds all_invs) s"
lunit'_in_hom: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and T :: "'b \<times> 'b \<Rightarrow> 'b"   and \<alpha> :: "'b \<times> 'b \<times> 'b \<Rightarrow> 'b"   and \<iota> :: "'b"   and a :: "'b" assumes "monoidal_category C T \<alpha> \<iota>"   and "partial_magma.ide C a" shows "partial_magma.in_hom C (category.inv C (monoidal_category.lunit C T \<alpha> \<iota> a)) a (T (monoidal_category.unity C \<iota>, a))"
fake_expanding: fixes H :: "chmsg set"   and T :: "atom set"   and U :: "fid_t set" shows "H \<subseteq> fake T U H"
partList0_partList1_equi: fixes ts :: "'b set list"   and s :: "'b set" assumes "disjoint_list_rec ts" shows "set (partList0 s ts) = set (partList1 s ts)"
x2: shows "x2 = (- (v\<^sub>o - v\<^sub>e) - sqrt D2) / (a\<^sub>o - a\<^sub>e)"
lcs_red_single_fst_crit_pair: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "p \<noteq> 0"   and "component_of_term (lt p) = component_of_term (lt q)" shows "red_single (monomial (- (1::'c)) (term_of_pair (lcs (lp p) (lp q), component_of_term (lt p)))) (fst (crit_pair p q)) p (lcs (lp p) (lp q) - lp p)"
empty_sets: fixes M :: "'d measure" shows "{} \<in> sets M"
sup_complement: fixes x :: "'a" shows "- x \<squnion> - - x = top"
iso_unit: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "horizontal_homs.obj V src a" shows "category.iso V (\<i> a)"
slsd_def: fixes x :: "'b \<times> 'c" shows "prod.slsd x = (times_prod' top_prod x = x)"
F_inf_def: fixes K :: "'a \<Rightarrow> 'a pmf set"   and S :: "'a set"   and S1 :: "'a set"   and S2 :: "'a set"   and f :: "'a \<Rightarrow> ennreal" assumes "Reachability_Problem K S S1 S2" shows "Reachability_Problem.F_inf K S S1 S2 f = (\<lambda>s\<in>S. if s \<in> S2 then 1 else if s \<in> S1 then \<Sqinter>D\<in>K s. integral\<^sup>N (measure_pmf D) f else 0)"
indspaceD: fixes f :: "('f, 'g) aezfun \<Rightarrow> 'v"   and G :: "'g set"   and FH :: "('f, 'g) aezfun set"   and V :: "'v set" assumes "f \<in> indspace G FH V" shows "f \<in> addfunset FH V \<inter> smultfunset G FH"
elims_2: fixes x :: "atom" assumes "is_Less x" obtains r :: "real" and rs :: "real list" where "x = Less r rs"
deg: shows "degree (poly_mod.Mp (m * k) f) = degree f"
finite_Collect_disjI: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" shows "finite {x. P x \<or> Q x} = (finite {x. P x} \<and> finite {x. Q x})"
SCF_R2: fixes xs :: "'c option list"   and P :: "'c option process"   and X :: "'c option set"   and Y :: "'c option set"   and Q :: "'c option process" assumes "xs \<in> sentences P"   and "(xs, X) \<in> failures P"   and "([], Y) \<in> failures Q" shows "(xs, insert None X \<inter> Y) \<in> seq_comp_failures P Q"
primitiveD: fixes f :: "'b poly"   and x :: "'b" assumes "primitive f"   and "\<And>y. y \<in> set (coeffs f) \<Longrightarrow> x dvd y" shows "x dvd (1::'b)"
simps_3: fixes w :: "nat \<Rightarrow> 'b"   and \<phi> :: "'b pltl"   and \<psi> :: "'b pltl" shows "w \<Turnstile>\<^sub>p \<phi> implies\<^sub>p \<psi> = (w \<Turnstile>\<^sub>p \<phi> \<longrightarrow> w \<Turnstile>\<^sub>p \<psi>)"
hyps: assumes "mod_type_class.to_nat i < k"   and "\<not> is_zero_row i A"   and "k \<le> nrows A" shows "Hermite_of_upt_row_i A k ass res $ j $ (LEAST n. A $ i $ n \<noteq> (0::'b)) \<in> range (res (Hermite_of_upt_row_i A k ass res $ i $ (LEAST n. A $ i $ n \<noteq> (0::'b))))"
Outside_def: fixes R :: "('d \<times> 'e) set"   and X :: "'d set" shows "R outside X = R - X \<times> Range R"
ccpo_Sup_least: fixes Sup :: "'b set \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and z :: "'b" assumes "class.ccpo Sup less_eq less"   and "Complete_Partial_Order.chain less_eq A"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq x z" shows "less_eq (Sup A) z"
subgroup_def: fixes H :: "'f set"   and G :: "('f, 'g) monoid_scheme" shows "subgroup H G \<equiv> (H \<subseteq> carrier G \<and> (\<forall>x y. x \<in> H \<longrightarrow> y \<in> H \<longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<in> H)) \<and> \<one>\<^bsub>G\<^esub> \<in> H \<and> (\<forall>x. x \<in> H \<longrightarrow> inv\<^bsub>G\<^esub> x \<in> H)"
multiset_inter_commute: fixes A :: "'k multiset"   and B :: "'k multiset" shows "A \<inter># B = B \<inter># A"
total: fixes b :: "'a"   and a :: "'a" assumes "(b, a) \<notin> R"   and "a \<in> set (x # xs)"   and "b \<in> set (x # xs)" shows "(a, b) \<in> R"
inverse_map_identity: fixes S :: "'a set" shows "inverse_map (identity S) S S = identity S"
stable_def: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and step :: "nat \<Rightarrow> 'a \<Rightarrow> (nat \<times> 'a) list"   and \<tau>s :: "'a list"   and p :: "nat" shows "stable r step \<tau>s p = (\<forall>(q, \<tau>)\<in>set (step p (\<tau>s ! p)). \<tau> \<sqsubseteq>\<^bsub>r\<^esub> \<tau>s ! q)"
distinct_step: fixes n1 :: "nat"   and signs1 :: "rat list list"   and n2 :: "nat"   and signs2 :: "rat list list" assumes "well_def_signs n1 signs1"   and "well_def_signs n2 signs2"   and "distinct signs1"   and "distinct signs2" shows "distinct (signs_smash signs1 signs2)"
devBC_act: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and cmp :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and pin :: "'b \<Rightarrow> 'a list set"   and pout :: "'b \<Rightarrow> 'a list"   and bc :: "'b \<Rightarrow> 'a list"   and mining :: "'b \<Rightarrow> bool"   and honest :: "'a \<Rightarrow> bool"   and cb :: "nat"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and nid :: "'a"   and n' :: "nat" assumes "Blockchain active cmp pin pout bc mining honest cb"   and "\<not> Option.is_none (Blockchain.devBC active cmp bc mining t n nid n')" shows "active (the (Blockchain.devBC active cmp bc mining t n nid n')) (t n')"
Montel: fixes S :: "complex set"   and \<H> :: "(complex \<Rightarrow> complex) set"   and \<F> :: "nat \<Rightarrow> complex \<Rightarrow> complex" assumes "open S"   and "\<And>h. h \<in> \<H> \<Longrightarrow> h holomorphic_on S"   and "\<And>K. \<lbrakk>compact K; K \<subseteq> S\<rbrakk> \<Longrightarrow> \<exists>B. \<forall>h\<in>\<H>. \<forall>z\<in>K. cmod (h z) \<le> B"   and "range \<F> \<subseteq> \<H>" obtains g :: "complex \<Rightarrow> complex" and r :: "nat \<Rightarrow> nat" where "g holomorphic_on S" and "strict_mono r" and "\<And>x. x \<in> S \<Longrightarrow> (\<lambda>n. \<F> (r n) x) \<longlonglongrightarrow> g x" and "\<And>K. \<lbrakk>compact K; K \<subseteq> S\<rbrakk> \<Longrightarrow> uniform_limit K (\<F> \<circ> r) g sequentially"
gt_sym_irrefl: fixes gt_sym :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a" assumes "gt_sym gt_sym" shows "\<not> gt_sym f f"
prems_1: shows "h \<turnstile> get_ancestors_si child \<rightarrow>\<^sub>r ancestors"
finite_field_factorization_main_uint32: fixes f :: "int poly"   and c :: "int"   and fs :: "int poly list" assumes "p \<le> 65535"   and "finite_field_factorization_main p (finite_field_ops32 (uint32_of_int p)) f = (c, fs)"   and "square_free_m f" shows "unique_factorization_m f (c, mset fs) \<and> c \<in> {0..<p} \<and> (\<forall>fi\<in>set fs. set (coeffs fi) \<subseteq> {0..<p})"
ereal_less_PInfty: fixes a :: "ereal" assumes "a \<noteq> \<infinity>" shows "a < \<infinity>"
Rep_Assertion: fixes x :: "'c Assertion" shows "Rep_Assertion x \<in> assertion"
Image_subsetI: fixes R :: "('a \<times> 'b) set"   and R' :: "('a \<times> 'b) set"   and X :: "'a set" assumes "R \<subseteq> R'" shows "R `` X \<subseteq> R' `` X"
adjacentset_def: fixes X :: "'b set set"   and x :: "'b set" assumes "SimplicialComplex X" shows "SimplicialComplex.adjacentset X x = {y \<in> X. x \<sim> y}"
axioms_1: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>S\<^sub>M\<^sub>C\<^sub>.\<^sub>f\<^sub>a\<^sub>i\<^sub>t\<^sub>h\<^sub>f\<^sub>u\<^sub>l\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>S\<^sub>M\<^sub>C\<^bsub>\<alpha>\<^esub> \<BB>"
totatives_prime: fixes p :: "nat" assumes "prime p" shows "totatives p = {0<..<p}"
equivE: fixes A :: "'g set"   and r :: "('g \<times> 'g) set" assumes "equiv A r" obtains "refl_on A r" and "sym r" and "trans r"
card_of_lists_cong: fixes A :: "'c set"   and B :: "'d set" assumes "|A| =o |B|" shows "|lists A| =o |lists B|"
mem_Sup_iff: fixes x :: "V"   and X :: "V set" shows "(x \<in> elts (\<Squnion> X)) = (x \<in> \<Union> (elts ` X) \<and> small X)"
neg_neg: fixes x :: "tr" shows "neg\<cdot>(neg\<cdot>x) = x"
elims: fixes x :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and xa :: "'a"   and xb :: "nat \<Rightarrow> 'b"   and xc :: "nat"   and y :: "'a \<times> 'b \<times> 'a" assumes "run\<^sub>t x xa xb xc = y" obtains \<delta> :: "'a \<Rightarrow> 'b \<Rightarrow> 'a" and q\<^sub>0 :: "'a" and w :: "nat \<Rightarrow> 'b" and i :: "nat" where "x = \<delta>" and "xa = q\<^sub>0" and "xb = w" and "xc = i" and "y = (run \<delta> q\<^sub>0 w i, w i, run \<delta> q\<^sub>0 w (Suc i))"
tuple_simps_2: fixes f :: "('a, 'a set) Maps.arr"   and g :: "('a, 'a set) Maps.arr"   and h :: "('a, 'a set) Maps.arr"   and k :: "('a, 'a set) Maps.arr" assumes "Maps.commutative_square f g h k" shows "Maps.dom (Maps.tuple h f g k) = Maps.dom h"
imirror_eq_nat_mirror_image: fixes I :: "nat set" shows "imirror I = (\<lambda>x. nat_mirror x (iMin I) (Max I)) ` I"
is_extensional: fixes f :: "'a" assumes "\<not> FG.A.arr f" shows "G' f = FG.B.null"
sdiv_minus_divisor: fixes a :: "int"   and b :: "int" shows "a sdiv - b = - (a sdiv b)"
cone_decomp_cone_subset: fixes T :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and hU :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set" assumes "cone_decomp T ps"   and "hU \<in> set ps" shows "cone hU \<subseteq> T"
abs_Un_eq: fixes G :: "msg set"   and H :: "msg set" shows "local.abs (G \<union> H) = local.abs G \<union> local.abs H"
product_mset_def: fixes A :: "'a multiset"   and B :: "'b multiset" shows "Multiset_More.product_mset A B = A \<times># B"
hensel_hypothesis: shows "eint 2 * val_Zp (f' \<bullet> a) < val_Zp (f \<bullet> a)"
of_int_diff: fixes i :: "int"   and j :: "int" shows "\<guillemotleft>i - j\<guillemotright> = \<guillemotleft>i\<guillemotright> \<ominus> \<guillemotleft>j\<guillemotright>"
plus_except: fixes p :: "'a \<Rightarrow>\<^sub>0 'b"   and t :: "'a" shows "p = Poly_Mapping.single t (lookup p t) + except p {t}"
rewrite_B2_cases: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and b :: "'b option"   and a :: "'a option" shows "(case a of None \<Rightarrow> True | Some s \<Rightarrow> case b of None \<Rightarrow> True | Some t \<Rightarrow> f s t) = (\<forall>s t. a = Some s \<and> b = Some t \<longrightarrow> f s t)"
assms_3: shows "\<^bold>|u\<^bold>| < \<^bold>|p\<^bold>|"
simps: fixes key_gen :: "('c \<times> 'd, 'a, 'b) gpv"   and aencrypt :: "'c \<Rightarrow> 'e \<Rightarrow> ('f, 'a, 'b) gpv"   and valid_plains :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and \<A>1 :: "'c \<Rightarrow> (('e \<times> 'e) \<times> 'g, 'a, 'b) gpv"   and \<A>2 :: "'f \<Rightarrow> 'g \<Rightarrow> (bool, 'a, 'b) gpv" shows "ind_cpa_pk.ind_cpa key_gen aencrypt valid_plains (\<A>1, \<A>2) = TRY key_gen \<bind> (\<lambda>(pk, sk). lift_spmf coin_spmf \<bind> (\<lambda>b. \<A>1 pk \<bind> (\<lambda>((m0, m1), \<sigma>). assert_gpv (valid_plains m0 m1) \<bind> (\<lambda>_. aencrypt pk (if b then m0 else m1) \<bind> (\<lambda>cipher. \<A>2 cipher \<sigma> \<bind> (\<lambda>guess. Generative_Probabilistic_Value.Done (guess = b))))))) ELSE lift_spmf coin_spmf"
inject: fixes x21 :: "('a, 'b) rbt"   and x22 :: "color"   and x23 :: "'a"   and x24 :: "'b"   and x25 :: "('a, 'b) rbt"   and y21 :: "('a, 'b) rbt"   and y22 :: "color"   and y23 :: "'a"   and y24 :: "'b"   and y25 :: "('a, 'b) rbt" shows "(Node x21 x22 x23 x24 x25 = Node y21 y22 y23 y24 y25) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23 \<and> x24 = y24 \<and> x25 = y25)"
star_sim3: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "z ; x = y ; z" shows "z ; x\<^sup>\<star> = y\<^sup>\<star> ; z"
assms: shows "norm (z - a) *\<^sub>R (b - z) \<noteq> norm (b - z) *\<^sub>R (z - a)"
lookup_eq_zero_in_keys_contradict: fixes f :: "'e \<Rightarrow>\<^sub>0 'd"   and k :: "'e" assumes "lookup f k = (0::'d)" shows "k \<notin> keys f"
in_hom_char: fixes f :: "(bool, 'arr list) concrete_category.arr"   and a :: "(bool, 'arr list) concrete_category.arr"   and b :: "(bool, 'arr list) concrete_category.arr" shows "\<guillemotleft>f : a \<rightarrow> b\<guillemotright> = (a = Zero \<and> b = Zero \<and> f = Zero \<or> a = One \<and> b = One \<and> f = One \<or> a = Zero \<and> b = One \<and> f \<in> fromArr ` Arr)"
list_length: fixes s :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "length (list s n) = n"
lookup_append_some: fixes env :: "('a, 'b, 'c) env"   and xs :: "'c list"   and e :: "('a, 'b, 'c) env"   and ys :: "'c list" assumes "lookup env xs = Some e" shows "lookup env (xs @ ys) = lookup e ys"
update_satisfies_tableau: fixes s :: "('i, 'a) state"   and x :: "nat"   and c :: "'a" assumes "\<triangle> (\<T> s)"   and "\<nabla> s"   and "x \<notin> lvars (\<T> s)" shows "\<langle>\<V> s\<rangle> \<Turnstile>\<^sub>t \<T> s \<longrightarrow> \<langle>\<V> (update x c s)\<rangle> \<Turnstile>\<^sub>t \<T> s"
not_sig_crit: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and ss :: "'t list"   and p :: "('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat"   and ps :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list"   and b :: "'t \<Rightarrow>\<^sub>0 'b" assumes "rb_aux_inv (bs, ss, p # ps)"   and "\<not> sig_crit bs (new_syz_sigs ss bs p) p"   and "b \<in> set bs" shows "lt b \<prec>\<^sub>t sig_of_pair p"
nb\<^sub>1: shows "\<forall>op'\<in>set ops. \<not> are_operators_interfering op op'"
prems: fixes x :: "'a" assumes "x \<in> insert x__ A" shows "(0::'b) \<le> f x"
inext_fix_iff: fixes n :: "nat"   and I :: "nat set" shows "(n \<notin> I \<or> finite I \<and> Max I = n) = (inext n I = n)"
empty_Mapping: shows "Mapping.empty = RBT_Mapping.Mapping RBT.empty"
seqE: fixes g :: "'a"   and f :: "'a" assumes "C.seq g f" obtains "C.arr f" and "C.arr g" and "C.dom g = C.cod f"
few_preds_redundant: fixes g :: "'g"   and \<phi> :: "'val" assumes "card (trueArgs g \<phi>) < 2"   and "unnecessaryPhi g \<phi>" shows "redundant_set g (reachable g \<phi>)"
Diff_mono: fixes A :: "'b set"   and C :: "'b set"   and D :: "'b set"   and B :: "'b set" assumes "order.greater_eq C A"   and "order.greater_eq B D" shows "order.greater_eq (C - D) (A - B)"
restriction_add_clm: fixes v :: "view"   and u :: "view"   and w :: "view"   and ts :: "traffic"   and c :: "cars" shows "v=u--w \<longrightarrow> |restrict v (clm ts) c| = |restrict u (clm ts) c| + |restrict w (clm ts) c|"
p_2: fixes K :: "'a set" assumes "K \<in> p" shows "K \<subseteq> cbox a b"
image_add_0: fixes S :: "'a set" shows "(+) (0::'a) ` S = S"
analz_Un_implConfidSet_aux_aux: fixes G :: "msg set"   and H :: "msg set"   and Ag :: "(agent \<times> agent) set" assumes "Enc_keys_clean (G \<union> H)" shows "analz (implConfidSet_aux Ag G \<union> H) \<subseteq> implConfidSet_aux Ag G \<union> PairSet AgentSet G \<union> synth (analz (G \<union> H))"
fls_regpart_of_nat: fixes n :: "nat" shows "fls_regpart (of_nat n) = of_nat n"
case_1: fixes f1 :: "'b"   and f2 :: "nat \<Rightarrow> 'b" shows "(case 0 of 0 \<Rightarrow> f1 | Suc x \<Rightarrow> f2 x) = f1"
path_singleton: fixes v :: "'a" assumes "v \<in> V" shows "path [v]"
\<L>\<^sub>P_wellformed_tokens: fixes p :: "(symbol \<times> character list) list" assumes "terminals p \<in> \<L>\<^sub>P" shows "wellformed_tokens p"
Schur_Im: fixes smult' :: "('b, 'a) aezfun \<Rightarrow> 'e \<Rightarrow> 'e"   and W :: "'e set"   and T :: "'c \<Rightarrow> 'e" assumes "IrrFinGroupRepresentation G smult' W"   and "GRepHom smult' T"   and "T ` V \<subseteq> W"   and "T ` V \<noteq> 0" shows "T ` V = W"
A: shows "M \<in> {M. digraph_map (with_proj G) M}"
of_nat_sum: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and f :: "'b \<Rightarrow> nat"   and A :: "'b set" assumes "class.semiring_1 one times plus zero" shows "semiring_1.of_nat one plus zero (sum f A) = comm_monoid_add.sum plus zero (\<lambda>x. semiring_1.of_nat one plus zero (f x)) A"
bitzero_def: fixes t :: "t" shows "bitzero t = (case t of T_i32 \<Rightarrow> ConstInt32 0 | T_i64 \<Rightarrow> ConstInt64 0 | T_f32 \<Rightarrow> ConstFloat32 0 | T_f64 \<Rightarrow> ConstFloat64 0)"
i_take_drop_append: fixes m :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "m \<le> n" shows "f \<Down> m @ f \<Down> n \<up> m = f \<Down> n"
zip_map2: fixes xs :: "'a list"   and f :: "'c \<Rightarrow> 'b"   and ys :: "'c list" shows "zip xs (map f ys) = map2 (\<lambda>x y. (x, f y)) xs ys"
dickson_less_def: fixes ord :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ord_strict :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and d :: "'a \<Rightarrow> nat"   and m :: "nat"   and s :: "'a"   and t :: "'a" assumes "gd_powerprod ord ord_strict" shows "gd_powerprod.dickson_less ord_strict d m s t = (d s \<le> m \<and> d t \<le> m \<and> ord_strict s t)"
Inl_in_Plus: fixes x :: "'c"   and A :: "'c set"   and B :: "'d set" shows "(Inl x \<in> A <+> B) = (x \<in> A)"
recurrent_subset_incl_1: fixes A :: "'a set" shows "recurrent_subset A \<subseteq> A"
f2_def: shows "f2 = (\<lambda>x. g2 x - C x)"
in_un1: fixes x :: "'c"   and A :: "'c set"   and B :: "'c set" assumes "x \<in> A \<union> B"   and "x \<notin> B" shows "x \<in> A"
dghm_singleton_components_2: fixes j :: "V"   and \<CC> :: "V" shows "dghm_singleton j \<CC>\<lparr>ArrMap\<rparr> = (\<lambda>f\<in>\<^sub>\<circ>\<CC>\<lparr>Arr\<rparr>. ZFC_in_HOL.set {\<langle>j, f\<rangle>})"
typing_case_Let: fixes \<Gamma> :: "(name \<times> ty) list"   and p :: "pat"   and t :: "trm"   and u :: "trm"   and U :: "ty" assumes "\<Gamma> \<turnstile> (LET p = t IN u) : U"   and "supp p \<sharp>* \<Gamma>"   and "distinct (pat_vars p)" obtains T :: "ty" and \<Delta> :: "(name \<times> ty) list" where "\<Gamma> \<turnstile> t : T" and "\<turnstile> p : T \<Rightarrow> \<Delta>" and "\<Delta> @ \<Gamma> \<turnstile> u : U"
Infinitesimal_def: shows "Infinitesimal = {x. \<forall>r\<in>\<real>. 0 < r \<longrightarrow> hnorm x < r}"
ennreal_ind: fixes A :: "'a set"   and x :: "'a" shows "ennreal (indicat_real A x) = indicator A x"
filterpositions2_def: fixes P :: "'a \<Rightarrow> bool"   and l :: "'a list" shows "filterpositions2 P l = concat (map (\<lambda>n. if P (l ! n) then [n] else []) [0..<length l])"
1: fixes q_in_bs :: "bool" assumes "(q_in_bs, p, q) \<in> set (np gs bs hs data)" shows "thesis"
gen_akiyama_tanigawa_row_0: fixes f :: "nat \<Rightarrow> real"   and l :: "nat"   and u :: "nat" shows "gen_akiyama_tanigawa_row f 0 l u = map f [l..<u]"
\<beta>\<^sub>g_def: shows "\<beta>\<^sub>g = \<a>[tab\<^sub>1 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> \<a>[tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (f\<^sub>0gh\<^sub>1.\<phi> \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f) \<cdot> \<a>[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (local.inv f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]"
simps: fixes l :: "('b \<times> bal) tree"   and a :: "'b"   and ba :: "bal"   and r :: "('b \<times> bal) tree" shows "split_max \<langle>l, (a, ba), r\<rangle> = (if r = \<langle>\<rangle> then (l, a) else let (r', a') = split_max r; t' = if decr r r' then balL l a ba r' else \<langle>l, (a, ba), r'\<rangle> in (t', a'))"
nat_le_eq_zle: fixes w :: "int"   and z :: "int" assumes "0 < w \<or> 0 \<le> z" shows "(nat w \<le> nat z) = (w \<le> z)"
get_ancestors_pure: fixes node_ptr :: "(_) object_ptr"   and h :: "(_) heap" shows "pure (get_ancestors_si node_ptr) h"
\<sigma>_5: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<sigma>)"
lookup_empty: fixes k :: "'c" shows "Mapping.lookup Mapping.empty k = None"
Vfrom: fixes a :: "V"   and i :: "V" shows "Vfrom a i = a \<squnion> (SUP j\<in>elts i. VPow (Vfrom a j))"
simps: fixes l1 :: "lit"   and l2 :: "lit" shows "insert_clause (Upair l1 l2) = insert_mapping (negate l1, l2) \<circ> insert_mapping (negate l2, l1)"
one_hom_1_def: fixes hom :: "'a \<Rightarrow> 'b" shows "one_hom_1 hom \<equiv> one_hom hom \<and> one_hom_1_axioms hom"
path_image_sym: fixes p :: "real \<Rightarrow> 'b"   and q :: "real \<Rightarrow> 'b" assumes "pathfinish p = pathstart q"   and "pathfinish q = pathstart p" shows "path_image (p +++ q) = path_image (q +++ p)"
interfree_lemma: fixes c :: "'b ann_com"   and s :: "'b"   and r :: "'b ann_com option"   and t :: "'b"   and Ts :: "('b ann_com option \<times> 'b set) list"   and i :: "nat"   and q :: "'b set" assumes "(Some c, s) -1\<rightarrow> (r, t)"   and "interfree Ts"   and "i < length Ts"   and "Ts ! i = (Some c, q)" shows "interfree (Ts[i := (r, q)])"
homotopic_paths_lid: fixes p :: "real \<Rightarrow> 'b"   and s :: "'b set" assumes "path p"   and "path_image p \<subseteq> s" shows "homotopic_paths s (linepath (pathstart p) (pathstart p) +++ p) p"
lowlink_more_cong: fixes G :: "('c, 'd) graph_rec_scheme"   and s :: "('e, 'f) tarjan_state_scheme"   and s' :: "('e, 'f) tarjan_state_scheme" assumes "Tarjan G"   and "state.more s = state.more s'" shows "lowlink s = lowlink s'"
jordan_nf_def: fixes A :: "'b mat"   and n_as :: "(nat \<times> 'b) list" shows "jordan_nf A n_as \<equiv> 0 \<notin> fst ` set n_as \<and> similar_mat A (jordan_matrix n_as)"
elims_1: fixes x :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xa :: "'c list"   and y :: "bool" assumes "successively x xa = y" obtains P :: "'c \<Rightarrow> 'c \<Rightarrow> bool" where "x = P__" and "xa = []" and "y"   | P :: "'c \<Rightarrow> 'c \<Rightarrow> bool" and xb :: "'c" where "x = Pa__" and "xa = [xb__]" and "y"   | P :: "'c \<Rightarrow> 'c \<Rightarrow> bool" and xb :: "'c" and ya :: "'c" and xs :: "'c list" where "x = P" and "xa = xb # ya # xs" and "y = (P xb ya \<and> successively P (ya # xs))"
that: fixes y :: "'a" assumes "x = p ^ multiplicity p x * y"   and "\<not> p dvd y" shows "thesis"
assms: shows "countable \<^bold>V\<^bsub>G\<^esub>"
sublist_append_back: fixes l1 :: "'a list"   and l2 :: "'a list" shows "subseq l1 (l2 @ l1)"
headn_0: fixes p :: "'b poly"   and n :: "nat"   and m :: "nat" assumes "isnpolyh p n"   and "m < n" shows "headn p m = p"
face_of_polytope_insert2: fixes S :: "'b set"   and a :: "'b"   and F :: "'b set" assumes "polytope S"   and "a \<notin> affine hull S"   and "F face_of S" shows "convex hull insert a F face_of convex hull insert a S"