hcju/isabelleps huggingface dataset:
elims: fixes x :: "const"   and y :: "int set" assumes "\<gamma>_const x = y" obtains n :: "int" where "x = Const n" and "y = {n}"   | "x = Any" and "y = UNIV"
bot_unique: fixes a :: "'c" shows "(top \<le> a) = (a = top)"
dom_m_fmupd: fixes k :: "'c"   and C :: "'d"   and N :: "('c, 'd) fmap" shows "dom_m (fmupd k C N) = add_mset k (remove1_mset k (dom_m N))"
axioms_1: fixes S :: "'b set"   and pls :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b" assumes "comm_monoid_add_ow S pls z" shows "ab_semigroup_add_ow S pls"
invar_butlast: fixes bq :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list"   and t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" assumes "SkewBinomialHeapStruc.invar (bq @ [t])" shows "SkewBinomialHeapStruc.invar bq"
Prefix0_is_CSP: fixes c :: "'d \<Rightarrow> 'c" shows "is_CSP_process (R (true \<turnstile> do c \<and> (\<lambda>(A, A'). alpha_rp.more A' = alpha_rp.more A)))"
mkKripke_kripke: fixes ws :: "'d set"   and rels :: "'b \<Rightarrow> ('d \<times> 'd) set"   and val :: "'d \<Rightarrow> 'c \<Rightarrow> bool" shows "kripke (mkKripke ws rels val)"
constant_on_closureI: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set" assumes "f constant_on S"   and "continuous_on (closure S) f" shows "f constant_on closure S"
arl_set_def: shows "arl_set \<equiv> \<lambda>(a, n) i x. Array.upd i x a \<bind> (\<lambda>a. return (a, n))"
iter_def: fixes nrandom :: "natural"   and sz :: "natural"   and seed :: "natural \<times> natural" shows "Random_Pred.iter nrandom sz seed = Random_Pred.iter' TYPE('e) nrandom sz seed"
return_pmf_card_equals_set: fixes S :: "'b set" shows "card {return_pmf x |x. x \<in> S} = card S"
node_deliver_messages_def: fixes history :: "nat \<Rightarrow> 'c event list"   and msg_id :: "'c \<Rightarrow> 'd"   and cs :: "'c event list" assumes "network history msg_id" shows "trivial_network.node_deliver_messages cs \<equiv> List.map_filter (\<lambda>e. case e of Broadcast x \<Rightarrow> Map.empty x | Deliver m \<Rightarrow> Some m) cs"
module_hom_ow_def: fixes U\<^sub>M\<^sub>_\<^sub>1 :: "'d set"   and plus\<^sub>M\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero\<^sub>M\<^sub>_\<^sub>1 :: "'d"   and minus\<^sub>M\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus\<^sub>M\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd"   and scale\<^sub>1 :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and U\<^sub>M\<^sub>_\<^sub>2 :: "'f set"   and plus\<^sub>M\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero\<^sub>M\<^sub>_\<^sub>2 :: "'f"   and minus\<^sub>M\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and uminus\<^sub>M\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f"   and scale\<^sub>2 :: "'e \<Rightarrow> 'f \<Rightarrow> 'f"   and f :: "'d \<Rightarrow> 'f" shows "module_hom_ow U\<^sub>M\<^sub>_\<^sub>1 plus\<^sub>M\<^sub>_\<^sub>1 zero\<^sub>M\<^sub>_\<^sub>1 minus\<^sub>M\<^sub>_\<^sub>1 uminus\<^sub>M\<^sub>_\<^sub>1 scale\<^sub>1 U\<^sub>M\<^sub>_\<^sub>2 plus\<^sub>M\<^sub>_\<^sub>2 zero\<^sub>M\<^sub>_\<^sub>2 minus\<^sub>M\<^sub>_\<^sub>2 uminus\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>2 f \<equiv> module_ow U\<^sub>M\<^sub>_\<^sub>1 plus\<^sub>M\<^sub>_\<^sub>1 zero\<^sub>M\<^sub>_\<^sub>1 minus\<^sub>M\<^sub>_\<^sub>1 uminus\<^sub>M\<^sub>_\<^sub>1 scale\<^sub>1 \<and> module_ow U\<^sub>M\<^sub>_\<^sub>2 plus\<^sub>M\<^sub>_\<^sub>2 zero\<^sub>M\<^sub>_\<^sub>2 minus\<^sub>M\<^sub>_\<^sub>2 uminus\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>2 \<and> module_hom_ow_axioms U\<^sub>M\<^sub>_\<^sub>1 plus\<^sub>M\<^sub>_\<^sub>1 scale\<^sub>1 U\<^sub>M\<^sub>_\<^sub>2 plus\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>2 f"
abs_inverse: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and equal :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and rep_nat :: "'a \<Rightarrow> nat"   and abs_nat :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "class.nat minus plus zero equal less_eq less rep_nat abs_nat" shows "rep_nat (abs_nat n) = n"
set_intros_15: fixes ym :: "'b"   and x122 :: "'b ltlc"   and x121 :: "'b ltlc" assumes "ym \<in> atoms_ltlc x122" shows "ym \<in> atoms_ltlc (x121 R\<^sub>c x122)"
int_plus: fixes n :: "nat"   and m :: "nat" shows "int (n + m) = int n + int m"
groupF_nequality: fixes A :: "'a list"   and f :: "'a \<Rightarrow> 'b"   and xs :: "'a list"   and B :: "'a list" assumes "A \<in> set (groupF f xs)"   and "B \<in> set (groupF f xs)"   and "A \<noteq> B" shows "\<forall>a\<in>set A. \<forall>b\<in>set B. f a \<noteq> f b"
assms_4: shows "(state0, state) \<in> transitionRelation F0 decisionVars"
integral_divide: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> 'b"   and y :: "'b" assumes "integrable M f" shows "LINT x|M. f x / y = integral\<^sup>L M f / y"
P1: shows "(LEAST n. A $ n $ j \<noteq> (0::'a) \<and> i \<le> n) = n"
finite_fset: fixes S :: "'b fset" shows "finite \<langle>S\<rangle>\<^sub>f"
simps_3: fixes v :: "'b tree"   and va :: "'b" shows "pass\<^sub>1 \<langle>v, va, \<langle>\<rangle>\<rangle> = \<langle>v, va, \<langle>\<rangle>\<rangle>"
rep_props_5: fixes r :: "'a" assumes "ide r" shows "tab\<^sub>1 r \<star> (tab\<^sub>0 r)\<^sup>* \<cong> r"
in_set_image_conv_nth: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b"   and l :: "'b list" shows "(f x \<in> f ` set l) = (\<exists>i<length l. f (l ! i) = f x)"
contracted_no_degree2_simp: fixes H :: "'a pair_pre_digraph" assumes "subdivision_pair G H"   and "verts3 (with_proj G) = pverts G" shows "gen_contr_graph (with_proj H) (verts3 (with_proj H)) = G"
lub_is_either: fixes S :: "'b ttree set" shows "lub S = Either S"
sum_add: fixes A :: "'b set"   and B :: "'b set"   and f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a" assumes "finite A"   and "finite B" shows "sum f A + sum g B = sum f (A - B) + sum g (B - A) + (\<Sum>x\<in>A \<inter> B. f x + g x)"
mem_convex_alt: fixes S :: "'a set"   and x :: "'a"   and y :: "'a"   and u :: "real"   and v :: "real" assumes "convex S"   and "x \<in> S"   and "y \<in> S"   and "0 \<le> u"   and "0 \<le> v"   and "0 < u + v" shows "(u / (u + v)) *\<^sub>R x + (v / (u + v)) *\<^sub>R y \<in> S"
bounded_lattice_def: fixes L :: "('c, 'd) gorder_scheme" shows "bounded_lattice L \<equiv> lattice L \<and> weak_partial_order_bottom L \<and> weak_partial_order_top L"
fixed_point_imp_closed_orbit_period_zero_1: fixes x :: "'a" assumes "x \<in> X"   and "f x = (0::'a)" shows "closed_orbit x"
take_bit_eq_mask_iff_exp_dvd: fixes n :: "nat"   and k :: "int" shows "(take_bit n k = mask n) = (2 ^ n dvd k + 1)"
connected_components_of_eq_singleton: fixes X :: "'b topology"   and S :: "'b set" shows "(connected_components_of X = {S}) = (connected_space X \<and> topspace X \<noteq> {} \<and> S = topspace X)"
eq_add_iff2: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and e :: "'b"   and c :: "'b"   and b :: "'b"   and d :: "'b" assumes "class.ring plus zero minus uminus times" shows "(plus (times a e) c = plus (times b e) d) = (c = plus (times (minus b a) e) d)"
lexordp_simps_1: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list" shows "ord.lexordp less [] ys = (ys \<noteq> [])"
rsinf: shows "r @@ s \<in> A\<^sup>\<omega>"
rep_eq: fixes x :: "'b \<Rightarrow> 'a"   and xa :: "'b fset" shows "fsum x xa = sum x (fset xa)"
vsubst_preserves_good: fixes Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and xs :: "'varSort"   and x1 :: "'var"   and x :: "'var" assumes "good Y" shows "good (Y #[x1 // x]_xs)"
ZF2mor_def: fixes C :: "('e, 'd, 'f) LSCategory_scheme"   and f :: "ZF" shows "z2m\<^bsub>C\<^esub>f \<equiv> THE m. m \<in> mor\<^bsub>C\<^esub> \<and> m2z\<^bsub>C\<^esub>m = f"
case_2: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'c \<Rightarrow> 'a"   and x2 :: "'c" shows "(case Inr x2 of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = f2 x2"
has_integral_bound_real': fixes B :: "real"   and f :: "real \<Rightarrow> 'b"   and i :: "'b"   and a :: "real"   and b :: "real"   and s :: "real set" assumes "0 \<le> B"   and "(f has_integral i) (cbox a b)"   and "finite s"   and "\<forall>x\<in>cbox a b - s. norm (f x) \<le> B" shows "norm i \<le> B * content {a..b}"
indRelLTPO_impl_TRel_is_weak_barbed_simulation: fixes TRel :: "('procT \<times> 'procT) set" assumes "weak_barbed_simulation (indRelLTPO TRel) (STCalWB SWB TWB)" shows "weak_barbed_simulation (TRel\<^sup>+) TWB"
sum_closed: fixes X :: "'b set"   and f :: "'b \<Rightarrow> 'a" assumes "\<And>x. x \<in> X \<Longrightarrow> f x \<in> A" shows "sum f X \<in> A"
\<Psi>_def: shows "\<Psi> = \<lparr>edge = edge', capacity = cap, source = SOURCE, sink = vertex.SINK\<rparr>"
assms_2: fixes u :: "'a"   and v :: "'a" assumes "u \<in> verts G"   and "v \<in> verts G" shows "u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v"
l_create_element\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_create_element\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_tag_name set_tag_name_locs DocumentClass.type_wf create_element DocumentClass.known_ptr"
ind_meas: shows "indicat_real A \<in> borel_measurable Invariants"
cadd_def: fixes \<kappa> :: "V"   and \<mu> :: "V" shows "\<kappa> \<oplus> \<mu> \<equiv> vcard (\<kappa> \<Uplus> \<mu>)"
stutter_extend_wf: fixes E :: "('a \<times> 'a) set"   and V :: "'a set" assumes "E \<subseteq> V \<times> V" shows "stutter_extend_edges V E \<subseteq> V \<times> V"
inject: fixes x21 :: "'c"   and x22 :: "'c bt"   and x23 :: "'c bt"   and y21 :: "'c"   and y22 :: "'c bt"   and y23 :: "'c bt" shows "(Br x21 x22 x23 = Br y21 y22 y23) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23)"
mem_set_ar_cos: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set"   and a :: "'c" assumes "Ring R"   and "ideal R I"   and "a \<in> carrier R" shows "a \<uplus>\<^bsub>R\<^esub> I \<in> set_ar_cos R I"
integrable_mult_left_iff: fixes M :: "'c measure"   and c :: "real"   and f :: "'c \<Rightarrow> real" shows "integrable M (\<lambda>x. c * f x) = (c = 0 \<or> integrable M f)"
writerT_below_iff: fixes a :: "'e\<cdot>('f, 'g) writerT"   and b :: "'e\<cdot>('f, 'g) writerT" shows "(a \<sqsubseteq> b) = (runWriterT\<cdot>a \<sqsubseteq> runWriterT\<cdot>b)"
lists_typings: fixes e :: "nat \<Rightarrow> LambdaType.type"   and ts :: "dB list"   and Ts :: "LambdaType.type list" assumes "e \<tturnstile> ts : Ts" shows "listsp (\<lambda>t. \<exists>T. e \<turnstile> t : T) ts"
real_alg_2_def: fixes rai :: "int poly \<times> rat \<times> rat" shows "real_alg_2 rai \<equiv> let p = poly_real_alg_1 rai in if degree p = 1 then Rational (Rat.Fract (- coeff p 0) (coeff p 1)) else case normalize_bounds_1 rai of (p', l, r) \<Rightarrow> Irrational (root_info.number_root (root_info p) r) (p', l, r)"
PO_s0g_secrecy: shows "reach s0g \<subseteq> s0g_secrecy"
dom_CompFun: fixes HA :: "('a, 'b, 'c) hierauto" shows "dom (CompFun HA) = HAStates HA"
lossless_def: shows "ind_cpa.lossless = (\<lambda>(\<A>1, \<A>2). (\<forall>pk. lossless_gpv \<I>_full (\<A>1 pk)) \<and> (\<forall>cipher \<sigma>. lossless_gpv \<I>_full (\<A>2 cipher \<sigma>)))"
rep_eq: fixes x :: "'h \<Rightarrow> ('f, 'g) generator \<times> 'g"   and xa :: "('h, 'e) generator" shows "generator (maps_trans x xa) = maps_raw x (generator xa)"
simps_2: fixes var :: "nat"   and p :: "real mpoly" shows "substNegInfinity var (Less p) = alternateNegInfinity p var"
equivalence_map_def: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data" shows "Span.equivalence_map f \<equiv> \<exists>g \<eta> \<epsilon>. Span.internal_equivalence f g \<eta> \<epsilon>"
hull_minimal: fixes s :: "'a set"   and t :: "'a set"   and S :: "'a set \<Rightarrow> bool" assumes "s \<subseteq> t"   and "S t" shows "S hull s \<subseteq> t"
wens_single_finite_Suc_eq_wens: fixes act :: "('b \<times> 'b) set"   and B :: "'b set"   and k :: "nat"   and init :: "'b set"   and allowed :: "('b \<times> 'b) set set" assumes "single_valued act" shows "wens_single_finite act B (Suc k) = wens (mk_program (init, {act}, allowed)) act (wens_single_finite act B k)"
comp_in_homI: fixes f :: "'a"   and a :: "'a"   and b :: "'a"   and g :: "'a"   and c :: "'a" assumes "Right.in_hom f a b"   and "Right.in_hom g b c" shows "Right.in_hom (Right.comp g f) a c"
refl_onD: fixes A :: "'e set"   and r :: "('e \<times> 'e) set"   and a :: "'e" assumes "refl_on A r"   and "a \<in> A" shows "(a, a) \<in> r"
HEndPhase0_valueChosen2: fixes s :: "state"   and s' :: "state"   and q :: "Proc"   and D :: "Disk set"   and b :: "nat"   and p :: "Proc" assumes "HEndPhase0 s s' q"   and "\<forall>d\<in>D. b \<le> bal (disk s d p) \<and> (\<forall>q. phase s q = 1 \<and> b \<le> mbal (dblock s q) \<and> hasRead s q d p \<longrightarrow> (\<exists>br\<in>blocksRead s q d. b \<le> bal (block br)))" shows "\<forall>d\<in>D. b \<le> bal (disk s' d p) \<and> (\<forall>q. phase s' q = 1 \<and> b \<le> mbal (dblock s' q) \<and> hasRead s' q d p \<longrightarrow> (\<exists>br\<in>blocksRead s' q d. b \<le> bal (block br)))"
del_order: fixes k :: "nat"   and t :: "'a btree"   and x :: "'a" assumes "0 < k"   and "root_order k t"   and "bal t" shows "almost_order k (del k x t)"
imgSbs_def: shows "imgSbs = prod_emb UNIV PM {0..n - 1} (\<Pi>\<^sub>E i\<in>{0..n - 1}. {x !! i})"
mcont_contD: fixes luba :: "'c set \<Rightarrow> 'c"   and orda :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and lubb :: "'d set \<Rightarrow> 'd"   and ordb :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd"   and Y :: "'c set" assumes "mcont luba orda lubb ordb f"   and "Complete_Partial_Order.chain orda Y"   and "Y \<noteq> {}" shows "f (luba Y) = lubb (f ` Y)"
Length_0_inj: fixes \<AA> :: "'i"   and \<BB> :: "'i" assumes "Length \<AA> = 0"   and "Length \<BB> = 0"   and "#\<^sub>V \<AA> = #\<^sub>V \<BB>" shows "\<AA> = \<BB>"
le_top: fixes x :: "'a" shows "x \<sqsubseteq>\<^bsub>r\<^esub> \<top>"
hom_mult: fixes x :: "'a poly"   and y :: "'a poly"   and a :: "'a" shows "poly (x * y) a = poly x a * poly y a"
List_unique1: fixes h :: "'c \<Rightarrow> 'c"   and p :: "'c"   and as :: "'c list" assumes "List h p as" shows "\<exists>!as. List h p as"
unfold_lesub_list: fixes xs :: "'b list"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list" shows "(xs [\<sqsubseteq>\<^bsub>r\<^esub>] ys) = Listn.le r xs ys"
prems_3: shows "fdisjnt (freess (z # zs)) (freess ys)"
swap_id_idempotent: fixes a :: "'a"   and b :: "'a" shows "Fun.swap a b id \<circ> Fun.swap a b id = id"
cases_scheme: fixes r :: "('b, 'c) arrow_of_spans_data_scheme" obtains Chn :: "'b" and Dom :: "'b span_data" and Cod :: "'b span_data" and more :: "'c" where "r = \<lparr>Chn = Chn, Dom = Dom, Cod = Cod, \<dots> = more\<rparr>"
piecewise_C1_differentiable_const: fixes c :: "'a"   and s :: "real set" shows "(\<lambda>x. c) piecewise_C1_differentiable_on s"
lift_Use_node: fixes V :: "'f"   and Use :: "'g \<Rightarrow> 'f set"   and n :: "'g"   and E :: "'g"   and X :: "'g"   and H :: "'f set"   and L :: "'f set" assumes "V \<in> Use n" shows "(Node n, V) \<in> lift_Use_set Use E X H L"
assms_3: fixes e :: "'a uprod" assumes "e \<in> F" shows "proper_uprod e"
inessential_eq_continuous_logarithm_circle: fixes S :: "'a set"   and f :: "'a \<Rightarrow> complex" shows "(\<exists>a. homotopic_with_canon (\<lambda>h. True) S (sphere 0 1) f (\<lambda>t. a)) = (\<exists>g. continuous_on S g \<and> (\<forall>x\<in>S. f x = exp (\<i> * complex_of_real (g x))))"
nonzero_inverse_mult_distrib: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "a \<noteq> zero"   and "b \<noteq> zero" shows "inverse (times a b) = times (inverse b) (inverse a)"
pred_stream_valid_cfg: fixes cfg :: "'s cfg"   and xs :: "'s cfg stream" assumes "cfg \<in> valid_cfg"   and "MC.enabled cfg xs" shows "pred_stream (\<lambda>cfg. cfg \<in> valid_cfg) xs"
rep_eq: fixes x :: "'c set_dlist" shows "DList_Set.length x = List.length (list_of_dlist x)"
p'_def: shows "p' = smult (inverse (lead_coeff p)) p"
even_card_loops: fixes w :: "'a" shows "even (card (in_arcs G w \<inter> out_arcs G w))"
trans: fixes Enc :: "'a \<Rightarrow> 'b"   and P :: "('a, 'b) Proc"   and Q :: "('a, 'b) Proc"   and R :: "('a, 'b) Proc" assumes "encoding.indRelRPOp Enc P Q"   and "encoding.indRelRPOp Enc Q R" shows "encoding.indRelRPOp Enc P R"
noether_nabla: fixes x :: "'a" assumes "Noetherian x" shows "\<nabla>x = (zero_class.zero::'a)"
dual_prod_def: fixes x :: "'c \<times> 'd" shows "dual_prod x = (\<partial> (fst x), \<partial> (snd x))"
emeasure_space_1: shows "emeasure M (space M) = 1"
hyps: shows "perm \<bullet> Q \<longmapsto> \<alpha> \<prec> Q'"
ritem_in_rset: fixes r :: "'b tree" assumes "r \<noteq> ET" shows "rightmost_item r \<in> set_of r"
B: shows "(\<delta> OclAllInstances_generic pre_post H) \<tau> = true \<tau>"
Basic_ntran: fixes f :: "'b \<Rightarrow> 'b"   and s :: "'b"   and Ts :: "('b ann_com option \<times> 'b set) list"   and t :: "'b"   and n :: "nat" assumes "(Basic f, s) -Pn\<rightarrow> (Parallel Ts, t)"   and "All_None Ts" shows "t = f s"
ys_def: shows "ys = vfsequence_of_vlist xs"
finite_dimensional_vector_space_pair_ow_def: fixes U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'f set"   and plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'f"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'f \<Rightarrow> 'f"   and scale\<^sub>1 :: "'g \<Rightarrow> 'f \<Rightarrow> 'f"   and basis\<^sub>1 :: "'f set"   and U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'h set"   and plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'h"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'h \<Rightarrow> 'h"   and scale\<^sub>2 :: "'g \<Rightarrow> 'h \<Rightarrow> 'h"   and basis\<^sub>2 :: "'h set" shows "finite_dimensional_vector_space_pair_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 basis\<^sub>1 U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 scale\<^sub>2 basis\<^sub>2 \<equiv> finite_dimensional_vector_space_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 basis\<^sub>1 \<and> finite_dimensional_vector_space_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 scale\<^sub>2 basis\<^sub>2"
sup_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> (y \<squnion> z) = x \<squnion> y \<squnion> z"
chine_assoc_props_4: shows "Prj\<^sub>0\<^sub>0 \<cdot> chine_assoc = Prj\<^sub>0"
eq_iff: fixes x :: "int"   and y :: "int" shows "(of_int x = of_int y) = (x = y)"
trivial_code_def: fixes v :: "'val"   and vs :: "'val list" shows "uninst_code.ssa.trivial_code v vs = (the_trivial v vs \<noteq> None)"
stutter_sampler_interval: fixes f :: "nat \<Rightarrow> nat"   and \<sigma> :: "nat \<Rightarrow> 'b"   and n :: "nat" assumes "stutter_sampler f \<sigma>" obtains k :: "nat" where "f k \<le> n" and "n < f (Suc k)"
prime_elem_not_unit: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and p :: "'a" assumes "class.comm_semiring_1 times one plus zero"   and "comm_semiring_1.prime_elem times one zero p" shows "\<not> dvd.dvd times p one"
prems_3: shows "\<forall>c\<in>set (c # cs). v c \<le> n"
stails_sfoldl': fixes xs :: "[:'a:]" shows "stails\<cdot>xs = sfoldl\<cdot>(\<Lambda> xs x. smap\<cdot>(\<Lambda> ys. ys :@ [:x:])\<cdot>xs :@ [:[::]:])\<cdot>[:[::]:]\<cdot>xs"
exact_decompD_2: fixes m :: "nat"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and U :: "'x set" assumes "exact_decomp m ps"   and "(h, U) \<in> set ps" shows "U \<subseteq> X"
set_ins_axioms: shows "set_ins rs.\<alpha> rs.invar rs.ins"
cont: fixes n :: "nat" shows "continuous_on {a..b} (\<lambda>x. if x \<le> a + 1 / real (Suc n) then real (Suc n) * d * (a + 1 / real (Suc n) - x) else 0)"
prod: shows "p = smult c (\<Prod>(a, i)\<leftarrow>bs. a ^ Suc i)"
ipset_from_cidr_not_empty: fixes base :: "'b word"   and len :: "nat" shows "ipset_from_cidr base len \<noteq> {}"
a_f_3_in_f: fixes s :: "'v"   and r :: "'v" assumes "(s, r) \<in> f" shows "\<not> P (nP s) s (nP r) r"
subdg_arrD: fixes f :: "V" assumes "f \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>" shows "f \<in>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>"
B: shows "four_block_mat A2 A1 A4 A3 \<in> carrier_mat (Suc n) (Suc n)"
intro: fixes G :: "('f, 'g) monoid_scheme"   and H :: "('h, 'i) monoid_scheme"   and h :: "'f \<Rightarrow> 'h" assumes "Group.group G"   and "Group.group H"   and "group_hom_axioms G H h" shows "group_hom G H h"
simple_function_borel_measurable: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure" assumes "f \<in> borel_measurable M"   and "finite (f ` space M)" shows "simple_function M f"
Gramian_determinant_1: fixes fs :: "int vec list"   and k :: "nat" assumes "LLL_invariant_weak fs"   and "k \<le> m" shows "rat_of_int (gs.Gramian_determinant fs k) = (\<Prod>j<k. \<parallel>gram_schmidt_fs.gso n (map of_int_hom.vec_hom fs) j\<parallel>\<^sup>2)"
vconn_implies_hpar_4: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and \<mu> :: "'c"   and f :: "'c"   and g :: "'c" assumes "horizontal_homs V src trg"   and "partial_magma.in_hom V \<mu> f g" shows "trg g = trg f"
inject_1: fixes x21 :: "'b"   and x22 :: "'a"   and y21 :: "'b"   and y22 :: "'a" shows "(x21:=x22 = y21:=y22) = (x21 = y21 \<and> x22 = y22)"
nz: shows "\<forall>\<^sub>F n in sequentially. f n \<noteq> (0::'a)"
elims_3: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and intT :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> 'd list \<Rightarrow> 'd"   and intP :: "'b \<Rightarrow> 'd list \<Rightarrow> bool"   and x :: "Preliminaries.var \<Rightarrow> 'd"   and xa :: "('a, 'b) lit" assumes "M.Struct wtFsym wtPsym arOf resOf intT intF intP"   and "\<not> Struct.satL intF intP x xa" obtains \<xi> :: "Preliminaries.var \<Rightarrow> 'd" and a :: "('a, 'b) atm" where "x = \<xi>__" and "xa = Pos a__" and "\<not> Struct.satA intF intP \<xi>__ a__"   | \<xi> :: "Preliminaries.var \<Rightarrow> 'd" and a :: "('a, 'b) atm" where "x = \<xi>" and "xa = Neg a" and "\<not> \<not> Struct.satA intF intP \<xi> a"
proj_poly_syz_def: fixes n :: "nat"   and b :: "'t \<Rightarrow>\<^sub>0 'c" shows "proj_poly_syz n b = Poly_Mapping.map_key (map_component (\<lambda>k. k + n)) b"
equiv_imp_le: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and u :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "equiv_trans t u" shows "le_trans t u"
Func_is_emp: fixes A :: "'d set"   and B :: "'e set" shows "(Func A B = {}) = (A \<noteq> {} \<and> B = {})"