hcju/isabelleps huggingface dataset:
rec_nat_Suc_imp: fixes f :: "nat \<Rightarrow> 'a"   and f1 :: "'a"   and f2 :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and n :: "nat" assumes "f = rec_nat f1 f2" shows "f (Suc n) = f2 n (f n)"
dgrad_p_set_le_def: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and G :: "('t \<Rightarrow>\<^sub>0 'c) set" shows "dgrad_p_set_le d F G = dgrad_set_le d (pp_of_term ` Keys F) (pp_of_term ` Keys G)"
connectedin_subset_topspace: fixes X :: "'c topology"   and S :: "'c set" assumes "connectedin X S" shows "S \<subseteq> topspace X"
mult_zero_ow_def: fixes U :: "'a set"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a" shows "mult_zero_ow U times zero \<equiv> times_ow U times \<and> zero_ow U zero \<and> mult_zero_ow_axioms U times zero"
inf_le2: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.semilattice_inf inf less_eq less" shows "less_eq (inf x y) y"
functor_FF: shows "functor VV.comp B.VV.comp DN.FF"
finite_imp_card_positive: fixes H :: "'a set"   and G :: "('a, 'b) monoid_scheme" assumes "subgroup H G"   and "finite (carrier G)" shows "0 < card H"
name_exists_fresh: fixes x :: "'b" obtains c :: "name" where "c \<sharp> x"
rev_nth_append: fixes xs :: "'b list"   and v :: "nat"   and y :: "'b"   and ys :: "'b list" assumes "xs !. v = Some y" shows "(ys @ xs) !. v = Some y"
extensional_restrict: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "f \<in> extensional A" shows "restrict f A = f"
less_eq_list_refl: fixes xs :: "'a list" shows "local.less_eq_list xs xs"
induced_morphism_def: fixes X :: "'d set"   and is_open\<^sub>X :: "'d set \<Rightarrow> bool"   and \<O>\<^sub>X :: "'d set \<Rightarrow> 'e set"   and \<rho>\<^sub>X :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str\<^sub>X :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str\<^sub>X :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str\<^sub>X :: "'d set \<Rightarrow> 'e"   and one_str\<^sub>X :: "'d set \<Rightarrow> 'e"   and Y :: "'f set"   and is_open\<^sub>Y :: "'f set \<Rightarrow> bool"   and \<O>\<^sub>Y :: "'f set \<Rightarrow> 'g set"   and \<rho>\<^sub>Y :: "'f set \<Rightarrow> 'f set \<Rightarrow> 'g \<Rightarrow> 'g"   and d :: "'g"   and add_str\<^sub>Y :: "'f set \<Rightarrow> 'g \<Rightarrow> 'g \<Rightarrow> 'g"   and mult_str\<^sub>Y :: "'f set \<Rightarrow> 'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero_str\<^sub>Y :: "'f set \<Rightarrow> 'g"   and one_str\<^sub>Y :: "'f set \<Rightarrow> 'g"   and f :: "'d \<Rightarrow> 'f"   and \<phi>\<^sub>f :: "'f set \<Rightarrow> 'g \<Rightarrow> 'e"   and x :: "'d" assumes "ind_mor_btw_stalks X is_open\<^sub>X \<O>\<^sub>X \<rho>\<^sub>X b add_str\<^sub>X mult_str\<^sub>X zero_str\<^sub>X one_str\<^sub>X Y is_open\<^sub>Y \<O>\<^sub>Y \<rho>\<^sub>Y d add_str\<^sub>Y mult_str\<^sub>Y zero_str\<^sub>Y one_str\<^sub>Y f \<phi>\<^sub>f x" shows "\<phi>\<^bsub>X is_open\<^sub>X \<O>\<^sub>X \<rho>\<^sub>X is_open\<^sub>Y \<O>\<^sub>Y \<rho>\<^sub>Y f \<phi>\<^sub>f x\<^esub> \<equiv> \<lambda>C\<in>stalk.carrier_stalk is_open\<^sub>Y \<O>\<^sub>Y \<rho>\<^sub>Y (f x). let r = SOME r. r \<in> C in direct_lim.class_of \<O>\<^sub>X \<rho>\<^sub>X {U. is_open\<^sub>X U \<and> x \<in> U} (f \<^sup>\<inverse> X (fst r)) (\<phi>\<^sub>f (fst r) (snd r))"
supp_name: fixes i :: "name" shows "supp i = {atom i}"
assms_2: shows "open S \<and> f x0 \<in> S"
preserves_dom: fixes f :: "'b" assumes "F.B.arr f" shows "G.B.dom (G f) = G (F.B.dom f)"
poly_rel_monic: shows "(pff.poly_rel ===> (=)) (monic_i (poly_ops ff_ops)) monic"
set_remove1_subset: fixes x :: "'l"   and xs :: "'l list" shows "set (remove1 x xs) \<subseteq> set xs"
Q1: shows "Quotient R1 Abs1 Rep1 T1"
prem: shows "fv (as @ \<Gamma>, body) \<subseteq> set L \<union> domA (as @ \<Gamma>)"
AAss_\<tau>red1t_xt1: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and a :: "'addr expr1"   and xs :: "'addr val list"   and a' :: "'addr expr1"   and xs' :: "'addr val list"   and i :: "'addr expr1"   and e :: "'addr expr1" assumes "\<tau>red1gt uf P t h (a, xs) (a', xs')" shows "\<tau>red1gt uf P t h (a\<lfloor>i\<rceil> := e, xs) (a'\<lfloor>i\<rceil> := e, xs')"
subset_fst_snd: fixes A :: "('b \<times> 'c) set" shows "A \<subseteq> fst ` A \<times> snd ` A"
eq_\<I>_gpv_coinduct_bind: fixes X :: "('h, 'i, 'j) gpv \<Rightarrow> ('k, 'i, 'j) gpv \<Rightarrow> bool"   and gpv :: "('h, 'i, 'j) gpv"   and gpv' :: "('k, 'i, 'j) gpv"   and A :: "'h \<Rightarrow> 'k \<Rightarrow> bool"   and \<I> :: "('i, 'j) \<I>" assumes "X gpv gpv'"   and "\<And>gpv gpv'. X gpv gpv' \<Longrightarrow> rel_spmf (eq_\<I>_generat A \<I> (\<lambda>gpv gpv'. X gpv gpv' \<or> eq_\<I>_gpv A \<I> gpv gpv' \<or> (\<exists>gpv'' gpv''' B f g. gpv = gpv'' \<bind> f \<and> gpv' = gpv''' \<bind> g \<and> eq_\<I>_gpv B \<I> gpv'' gpv''' \<and> rel_fun B X f g))) (the_gpv gpv) (the_gpv gpv')" shows "eq_\<I>_gpv A \<I> gpv gpv'"
cases: fixes x :: "('b, 'c) Combinators list" obtains xa :: "('b, 'c) Combinators" and xs :: "('b, 'c) Combinators list" where "x = xa # xs"   | "x = []"
map_sel_2: fixes f :: "'c \<Rightarrow> 'b"   and a :: "'c tree" shows "left (map_tree f a) = map_tree f (left a)"
d: shows "abc_steps_l (0, lma) B nb = (length B, lmb)"
integrable_zero: fixes M :: "'a measure" shows "integrable M (\<lambda>x. 0::'b)"
hs_done_locs_def: shows "hs_done_locs = (\<Union>l\<in>{''idle_noop'', ''idle_flip_noop'', ''init_noop'', ''mark_noop''} \<union> {''mark_loop_get_roots'', ''mark_loop_get_work''}. prefixed (l @ ''_done''))"
intro: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "antisymmetric A less_eq"   and "well_related_set A less_eq" shows "well_ordered_set A less_eq"
tendsto: fixes g :: "'c \<Rightarrow> 'a"   and a :: "'a"   and F :: "'c filter" assumes "(g \<longlongrightarrow> a) F" shows "((\<lambda>x. f (g x)) \<longlongrightarrow> f a) F"
rep_eq: fixes x :: "real"   and xa :: "'a \<Rightarrow>\<^sub>L 'c" shows "blinfun_apply (x *\<^sub>R xa) = (\<lambda>xb. x *\<^sub>R blinfun_apply xa xb)"
weight_subst: fixes t :: "('f, 'd) Term.term"   and \<sigma> :: "'d \<Rightarrow> ('f, 'c) Term.term" shows "weight (t \<cdot> \<sigma>) = weight t + (\<Sum>x\<in>#vars_term_ms (SCF t). weight (\<sigma> x) - w0)"
nonabort_j0: shows "chaos \<sqsubseteq> (j\<^sub>0::'a::refinement_lattice)"
path_eq_chain: shows "chain_reparam_weak_path \<gamma> one_chain"
OUT_h: fixes x :: "'v" shows "d_OUT h x = (if reachable x then d_OUT g x else 0)"
integral_pmf_of_set: fixes S :: "'c set"   and f :: "'c \<Rightarrow> real" assumes "S \<noteq> {}"   and "finite S" shows "measure_pmf.expectation (local.pmf_of_set S) f = sum f S / real (card S)"
multipl_div: fixes f :: "nat \<Rightarrow> complex"   and d1 :: "nat"   and m :: "nat"   and d2 :: "nat"   and k :: "nat" assumes "multiplicative_function f"   and "d1 dvd m"   and "d2 dvd k"   and "coprime m k" shows "f (m * k div (d1 * d2)) = f (m div d1) * f (k div d2)"
approx: shows "P,\<Phi> |- (None, h, (stk, loc, C, M, pc, ics) # frs, sh) [ok]"
m2r_rel_def: fixes r :: "('b \<times> 'b option) set" shows "m2r_rel r \<equiv> {(Some a, b) |a b. (a, b) \<in> r}"
reachable_target: fixes q :: "'c"   and A :: "('d, 'c) nba"   and p :: "'c"   and r :: "('d \<times> 'c) list" assumes "q \<in> NBA.reachable A p"   and "NBA.path A r q" shows "target r q \<in> NBA.reachable A p"
Unifier_in_unifiers_singleton: fixes s :: "('c, 'd) Term.term"   and \<theta> :: "'d \<Rightarrow> ('c, 'd) Term.term"   and t :: "('c, 'd) Term.term" shows "Unifier \<theta> s t = (\<theta> \<in> unifiers {(s, t)})"
f_Exec_Stream_expand_map_aggregate_Cons: fixes k :: "nat"   and f :: "'e \<Rightarrow> 'd"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and x :: "'f message_af"   and xs :: "'f message_af list"   and c :: "'e"   and ag :: "'d list \<Rightarrow> 'd" assumes "0 < k" shows "f_aggregate (map f (f_Exec_Comp_Stream trans_fun ((x # xs) \<odot> k) c)) k ag = ag (map f (f_Exec_Comp_Stream trans_fun (x # NoMsg\<^bsup>k - Suc 0\<^esup>) c)) # f_aggregate (map f (f_Exec_Comp_Stream trans_fun (xs \<odot> k) (f_Exec_Comp trans_fun (x # NoMsg\<^bsup>k - Suc 0\<^esup>) c))) k ag"
fs_int_mu_d_Z: fixes j :: "nat"   and ii :: "nat" assumes "j \<le> ii"   and "ii < length fs" shows "rat_of_int (fs.d fs (Suc j)) * fs.gs.\<mu> ii j \<in> \<int>"
subgroup_inverse_equality: fixes u :: "'a" assumes "u \<in> H" shows "inverse u = sub.inverse u"
getParts_ipPartition: fixes rs :: "'c simple_rule list" shows "ipPartition (set (map wordinterval_to_set (extract_IPSets rs))) (set (map wordinterval_to_set (getParts rs)))"
nth_less_subdegree_zero: fixes n :: "nat"   and f :: "'a fps" assumes "n < subdegree f" shows "fps_nth f n = (0::'a)"
lm079: shows "partitionValuedUniverse \<subseteq> Pow (UNIV \<times> (UNIV - {{}}))"
has_vderiv_on_If: fixes U :: "real set"   and S :: "real set"   and T :: "real set"   and f :: "real \<Rightarrow> 'a"   and f' :: "real \<Rightarrow> 'a"   and g :: "real \<Rightarrow> 'a"   and g' :: "real \<Rightarrow> 'a" assumes "U = S \<union> T"   and "(f has_vderiv_on f') (S \<union> closure T \<inter> closure S)"   and "(g has_vderiv_on g') (T \<union> closure T \<inter> closure S)"   and "\<And>x. \<lbrakk>x \<in> closure T; x \<in> closure S\<rbrakk> \<Longrightarrow> f x = g x"   and "\<And>x. \<lbrakk>x \<in> closure T; x \<in> closure S\<rbrakk> \<Longrightarrow> f' x = g' x" shows "((\<lambda>t. if t \<in> S then f t else g t) has_vderiv_on (\<lambda>t. if t \<in> S then f' t else g' t)) U"
subcls_direct2: fixes G :: "prog"   and C :: "qtname"   and c :: "class"   and D :: "qtname" assumes "class G C = Some c"   and "C \<noteq> Object"   and "D = super c" shows "G\<turnstile>C\<prec>\<^sub>C D"
asTerm_pick: fixes X :: "('c, 'd, 'varSort, 'var, 'e) qTerm set" assumes "good X" shows "asTerm (pick X) = X"
comp_def: fixes f :: "'d \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'd" shows "f \<circ> g = (\<lambda>x. f (g x))"
indifferent_pref_def: fixes x :: "'b"   and r :: "('b \<times> 'b) set"   and y :: "'b" shows "x \<^bsub>r\<^esub>\<approx> y \<equiv> x \<^bsub>r\<^esub>\<preceq> y \<and> y \<^bsub>r\<^esub>\<preceq> x"
mult_assoc_hyp: shows "matr_option (dim_row (M_mat signs subsets)) (mat_inverse (M_mat signs subsets)) *\<^sub>v (M_mat signs subsets *\<^sub>v w_vec p qs signs) = matr_option (dim_row (M_mat signs subsets)) (mat_inverse (M_mat signs subsets)) * M_mat signs subsets *\<^sub>v w_vec p qs signs"
lex_ext_def: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> bool \<times> bool"   and n :: "nat"   and ss :: "'b list"   and ts :: "'b list" shows "lex_ext f n ss ts = (let lts = length ts in if length ss = lts \<or> lts \<le> n then lex_ext_unbounded f ss ts else (False, False))"
conv_to_is_Nil_1: fixes l :: "'c list" shows "(l = []) = is_Nil l"
H: shows "(\<lambda>n. f (u n)) \<longlonglongrightarrow> f l"
sel_1: fixes x1 :: "real"   and x2 :: "real"   and x3 :: "real"   and x4 :: "real"   and x5 :: "real"   and x6 :: "real"   and x7 :: "real"   and x8 :: "real" shows "Ree (Octo x1 x2 x3 x4 x5 x6 x7 x8) = x1"
Vset_in_mono: fixes i :: "V"   and j :: "V" assumes "i \<in>\<^sub>\<circ> j" shows "Vset i \<in>\<^sub>\<circ> Vset j"
paths: fixes xs :: "'a list" assumes "xs \<in> paths" shows "v0 \<leadsto>xs\<leadsto>\<^bsub>remove_vertex v\<^esub> v1"
mds_consistent_def: fixes ev\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and ev\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and bexp_neg :: "'d \<Rightarrow> 'd"   and assign_post :: "'d set \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> 'd set"   and dma_type :: "'a \<Rightarrow> 'd set"   and pred_False :: "'d"   and mds :: "Mode \<Rightarrow> 'a set"   and \<Gamma> :: "'a \<Rightarrow> 'd set option"   and \<S> :: "'a set \<times> 'a set"   and P :: "'d set" assumes "sifum_types ev\<^sub>A ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars bexp_neg assign_post dma_type pred_False" shows "sifum_types.mds_consistent bexp_vars \<C>_vars mds \<Gamma> \<S> P \<equiv> \<S> = (mds AsmNoWrite, mds AsmNoReadOrWrite) \<and> dom \<Gamma> = {x. x \<notin> \<Union> (range \<C>_vars) \<and> sifum_lang_no_dma.stable \<S> x} \<and> (\<forall>p\<in>P. \<forall>x\<in>bexp_vars p. sifum_lang_no_dma.stable \<S> x)"
CSP2_def: fixes P :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" shows "CSP2 P \<equiv> P ;; J_csp"
intros: fixes x :: "ord0"   and y :: "ord0" assumes "\<forall>a. (a, x) \<in> ord0_prec\<^sup>+ \<longrightarrow> (\<exists>b. (b, y) \<in> ord0_prec\<^sup>+ \<and> (a, b) \<in> ord0_leq)" shows "(x, y) \<in> ord0_leq"
Abs_uexpr_inverse: fixes y :: "'c \<Rightarrow> 'd" assumes "y \<in> UNIV" shows "\<lbrakk>mk\<^sub>e y\<rbrakk>\<^sub>e = y"
brouwer: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'b" assumes "compact S"   and "convex S"   and "S \<noteq> {}"   and "continuous_on S f"   and "f ` S \<subseteq> S" obtains x :: "'b" where "x \<in> S" and "f x = x"
a_def: shows "a = 2829469759662002867886529831139137601191652261996513014734415222704732791803 / 1062141879292765061960538947347721564047051545995266466660439319087625011200"
ThrowThrow\<^sub>1: fixes P :: "nat exp prog"   and e :: "nat exp"   and s\<^sub>0 :: "heap \<times> val list \<times> sheap"   and e' :: "nat exp"   and s\<^sub>1 :: "heap \<times> val list \<times> sheap" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>throw e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>"
group_commutes_pow: fixes x :: "'a"   and y :: "'a"   and n :: "nat" assumes "x \<otimes> y = y \<otimes> x"   and "x \<in> carrier \<G>"   and "y \<in> carrier \<G>" shows "x [^] n \<otimes> y = y \<otimes> x [^] n"
empty_not_adjacent: fixes x :: "'b set" shows "\<not> {} \<sim> x"
fusion_part_character: fixes F :: "i \<Rightarrow> bool" assumes "GEM"   and "\<exists>x. F x" shows "\<forall>y. P y (\<sigma>v. F v) = (\<forall>w. P w y \<longrightarrow> (\<exists>v. F v \<and> O w v))"
7: fixes M :: "real" assumes "0 < M" shows "\<exists>\<^sub>\<infinity>k. (bb (k + 1) / aa (k + 1)) powr (\<epsilon> / (1 + \<epsilon>)) < 1 / (M powr (\<epsilon> / (1 + \<epsilon>)) * prod aa {0..k} powr (2 + \<delta> * \<epsilon> / (1 + \<epsilon>)))"
jb_imp_diff_ev: fixes A :: "'b mat"   and i :: "nat"   and j :: "nat" assumes "??.Jordan_Normal_Form_Existence.jb A i j" shows "??.Jordan_Normal_Form_Existence.diff_ev A i j"
components_are_iso: fixes a :: "'d" assumes "D.ide a" shows "C.iso (F a)"
simps_7: fixes x3 :: "nat"   and x1 :: "nat" shows "Regions.intv.Greater x3 \<noteq> Regions.intv.Const x1"
productive_onp_productive_on_eq: fixes g :: "'c \<Rightarrow> ('d, 'c) step" shows "productive_onp g = (\<lambda>x. x \<in> productive_on g)"
psubst_csubst: fixes a :: "coname"   and \<theta>_n :: "(name \<times> coname \<times> trm) list"   and \<theta>_c :: "(coname \<times> name \<times> trm) list"   and x :: "name"   and P :: "trm"   and M :: "trm" assumes "a \<sharp> (\<theta>_n, \<theta>_c)" shows "\<theta>_n,((a, x, P) # \<theta>_c)<M> = \<theta>_n,\<theta>_c<M>{a:=x.P}"
iNodeOf_tree_at: fixes fc :: "('form, 'rule) graph_node \<Rightarrow> nat \<Rightarrow> 'subst \<Rightarrow> 'form fset \<times> 'form \<Rightarrow> bool"   and t :: "('form, 'rule, 'subst, 'var) itnode rose_tree"   and ent :: "'form fset \<times> 'form"   and "is" :: "nat list" assumes "iwf fc t ent"   and "is \<in> it_paths t" shows "iNodeOf (tree_at t is) \<in> i.R nodes"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>J) (\<cdot>\<^sub>[\<^sub>A\<^sub>,\<^sub>B\<^sub>]) F G \<tau>"
maxsimpD_simplex: fixes x :: "'a set" assumes "maxsimp x" shows "x \<in> X"
in_existence_between_zeroI: fixes t :: "real"   and t0 :: "real"   and x0 :: "real"   and s :: "real" assumes "t \<in> existence_ivl t0 x0"   and "s \<in> {t0--t}" shows "s \<in> existence_ivl t0 x0"
integrable_sum: fixes I :: "'c set"   and M :: "'d measure"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e" assumes "\<And>i. i \<in> I \<Longrightarrow> integrable M (f i)" shows "integrable M (\<lambda>x. \<Sum>i\<in>I. f i x)"
rep_eq: fixes x :: "(String.literal \<times> event_data list) prefix" shows "msteps x = msteps0 (Rep_prefix x)"
lowEquivalence_def: fixes s :: "('var \<Rightarrow> 'val option) list"   and s' :: "('var \<Rightarrow> 'val option) list" shows "s \<approx>\<^sub>L s' \<equiv> \<forall>V\<in>L. hd s V = hd s' V"
ap_gpv_def: fixes f :: "('d \<Rightarrow> 'a, 'b, 'c) gpv"   and x :: "('d, 'b, 'c) gpv" shows "f \<diamondop> x = f \<bind> (\<lambda>f'. x \<bind> (\<lambda>x'. Generative_Probabilistic_Value.Done (f' x')))"
a_Inj2: fixes \<Gamma> :: "(Syntax.var, ty) fmap"   and e :: "Syntax.term"   and eP :: "Syntax.term"   and eV :: "Syntax.term"   and \<tau>\<^sub>2 :: "ty"   and \<tau>\<^sub>1 :: "ty" assumes "\<Gamma> \<turnstile> e, eP, eV : \<tau>\<^sub>2" shows "\<Gamma> \<turnstile> Inj2 e, Inj2 eP, Inj2 eV : Syntax.Sum \<tau>\<^sub>1 \<tau>\<^sub>2"
simps: fixes prec :: "nat"   and a :: "real"   and b :: "real"   and c :: "real"   and d :: "real"   and x :: "real" shows "intersect_segment_xline prec ((a, b), c, d) x = (if a \<le> c then intersect_segment_xline' prec ((a, b), c, d) x else intersect_segment_xline' prec ((c, d), a, b) x)"
evaluate_clock_monotone: fixes env :: "v sem_env"   and s :: "'b state"   and e :: "exp0" assumes "evaluate_dom (env, s, e)" shows "clock (fst (Evaluate_Single.evaluate env s e)) \<le> clock s"
reachable_imp_identifiers_finite_2: fixes s :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option" assumes "reachable s" shows "finite (LID\<^sub>G s)"
mann: fixes x :: "'a" shows "2 \<times> x = - (- 3 \<times> x \<squnion> x) \<squnion> 2 \<times> x"
dim_row_transfer_rule: fixes A :: "'d mat"   and A' :: "(('d, 'e) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec" assumes "HMA_M A A'" shows "dim_row A = CARD('f)"
abc_list_crsp_lm_v: fixes lma :: "nat list"   and lmb :: "nat list"   and n :: "nat" assumes "abc_list_crsp lma lmb" shows "abc_lm_v lma n = abc_lm_v lmb n"
Un_empty_right: fixes A :: "'g set" shows "A \<union> {} = A"
zero: shows "\<forall>x<length (IArray.list_of (vec_to_iarray A)). mod_type_class.to_nat i \<le> x \<longrightarrow> IArray.list_of (vec_to_iarray A) ! x = (0::'a)"
finite_distinct: fixes s :: "'b set" assumes "finite s" shows "finite {xs. set xs \<subseteq> s \<and> distinct xs}"
lookup_update_other: fixes n :: "'c"   and m :: "'c"   and destr :: "'e \<Rightarrow> 'd"   and destr' :: "'e \<Rightarrow> 'f"   and constr :: "'g \<Rightarrow> 'e"   and f :: "'f \<Rightarrow> 'g"   and s :: "'c \<Rightarrow> 'e" assumes "n \<noteq> m" shows "lookup destr n (update destr' constr m f s) = lookup destr n s"
dsafe_Diff_simps: fixes a :: "('a, 'b) trm" shows "dsafe (Differential a) = dfree a"
sdrop_while_sdrop_LEAST: fixes P :: "'b \<Rightarrow> bool"   and s :: "'b stream" assumes "\<exists>n. P (s !! n)" shows "sdrop_while (Not \<circ> P) s = sdrop (LEAST n. P (s !! n)) s"
exh_start: fixes p :: "'a" shows "p = \<bottom> \<or> (\<exists>x. p = x \<and> x \<noteq> \<bottom>)"
ntrunc_subsetI: fixes k :: "nat"   and M :: "('a, 'b) node set" shows "ntrunc k M \<subseteq> M"
Right_Coset_memE: fixes a :: "'a"   and H :: "'a set"   and x :: "'a" assumes "a \<in> H |\<cdot> x" obtains h :: "'a" where "h \<in> H" and "a = h \<cdot> x"
ssa_nth_inA: shows "\<forall>i<length ssa. ssa ! i \<in> A"
infinity_evalUni_LessUni: fixes p :: "real \<times> real \<times> real"   and x :: "real" shows "(\<exists>x. \<forall>y<x. aEvalUni (LessUni p) y) = evalUni (substNegInfinityUni (LessUni p)) x"
assms_2: fixes r :: "'a" assumes "r \<in> deps r2" shows "r \<noteq> i1"
similarE: fixes x :: "Value"   and y :: "CValue"   and n :: "nat" assumes "x \<triangleleft>\<triangleright> y" shows "\<psi>\<^sup>D\<^bsub>n\<^esub>\<cdot>x \<triangleleft>\<triangleright>\<^bsub>n\<^esub> \<psi>\<^sup>E\<^bsub>n\<^esub>\<cdot>y"
cnj_mult: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and cnj :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.cnj divide inverse times one plus zero minus uminus cnj" shows "cnj (times a b) = times (cnj a) (cnj b)"
insert_Nil: fixes x :: "'g" shows "List.insert x [] = [x]"
structInduct: fixes P :: "'c list \<Rightarrow> bool"   and l :: "'c list" assumes "P []"   and "\<forall>x xs. P xs \<longrightarrow> P (x # xs)" shows "P l"
nzero: shows "\<forall>x. c < x \<and> x \<le> d2 \<longrightarrow> (\<forall>q\<in>set (pders p). poly q x \<noteq> 0)"
assms_1: fixes x :: "'a"   and y :: "'a" assumes "R x y" shows "S (f x) (f y)"
inf_ccSUP: fixes B :: "'d set"   and a :: "'a"   and f :: "'d \<Rightarrow> 'a" assumes "countable B" shows "inf a (Sup (f ` B)) = (SUP b\<in>B. inf a (f b))"
homotopy_equivalence_reduced_homology_group_isomorphisms: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'a"   and p :: "int" assumes "continuous_map X Y f"   and "continuous_map Y X g"   and "homotopic_with (\<lambda>h. True) X X (g \<circ> f) id"   and "homotopic_with (\<lambda>k. True) Y Y (f \<circ> g) id" shows "group_isomorphisms (reduced_homology_group p X) (reduced_homology_group p Y) (hom_induced p X {} Y {} f) (hom_induced p Y {} X {} g)"
mult_inverse_of_int_commute: 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 xa :: "int"   and x :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "times (inverse (ring_1.of_int one plus zero minus xa)) x = times x (inverse (ring_1.of_int one plus zero minus xa))"
f1: shows "rk {a, c, A, C, R} \<le> 3"
castAbs_Abs_swap: fixes xs :: "'varSort"   and s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and y :: "'var"   and x :: "'var"   and x' :: "'var"   and X' :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set" assumes "isInBar (xs, s)"   and "wls s X"   and "y = x \<or> fresh xs y X"   and "y = x' \<or> fresh xs y X'"   and "castAbs xs s (Abs xs x X) = (x', X')" shows "X #[y \<and> x]_xs = X' #[y \<and> x']_xs"
prod_encode_def: shows "prod_encode = (\<lambda>(m, n). triangle (m + n) + m)"
seq: fixes p :: "assertion"   and c1 :: "command"   and q :: "assertion"   and c2 :: "command"   and r :: "assertion" assumes "prov_triple (p, c1, q)"   and "prov_triple (q, c2, r)" shows "prov_triple (p, c1 ;; c2, r)"
decseqD: fixes f :: "nat \<Rightarrow> 'b"   and i :: "nat"   and j :: "nat" assumes "decseq f"   and "i \<le> j" shows "f j \<le> f i"
openin_subtopology_Int: fixes X :: "'a topology"   and S :: "'a set"   and T :: "'a set" assumes "openin X S" shows "openin (subtopology X T) (S \<inter> T)"
mssnth_arctan_series_stream: fixes n :: "nat" shows "mssnth arctan_series_stream n = (- 1) ^ n / (2 * real n + 1)"