hcju/isabelleps huggingface dataset:
homeomorphic_imp_retraction_maps: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'c" assumes "homeomorphic_maps X Y f g" shows "retraction_maps X Y f g"
conv_dist_sup: fixes x :: "'a"   and y :: "'a" shows "id (x \<squnion> y) = id x \<squnion> id y"
empt_in: shows "{} \<in> Pow (b1 \<inter> b2)"
uminus_Inf: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and A :: "'d set" assumes "class.complete_boolean_algebra Inf Sup inf less_eq less sup bot top minus uminus" shows "uminus (Inf A) = Sup (uminus ` A)"
SeqT: fixes c1 :: "(test, atom) com"   and s :: "Concrete.var \<Rightarrow> nat"   and s' :: "Concrete.var \<Rightarrow> nat"   and c2 :: "(test, atom) com" assumes "Example_PL.transT (c1, s) s'" shows "Example_PL.transC (c1 ;; c2, s) (c2, s')"
star_slide_var1: fixes x :: "'a" shows "connected_root (x\<^sup>\<star>) x"
idem_star2: fixes s :: "'a"   and rs :: "('b \<times> 'c) list" shows "\<exists>rs'. s \<star> rs = s \<star> rs' \<and> set rs' \<subseteq> set rs \<and> (\<forall>r\<in>set rs'. \<not> contains s r)"
add_mset_remove_trivial_If: fixes a :: "'b"   and N :: "'b multiset" shows "add_mset a (N - {#a#}) = (if a \<in># N then N else add_mset a N)"
UpTo_aux: fixes d :: "'d"   and R :: "('com list \<times> 'com list) set"   and i :: "nat"   and V :: "'com list"   and V' :: "'com list"   and m1 :: "'id \<Rightarrow> 'val"   and W :: "'com list"   and m2 :: "'id \<Rightarrow> 'val"   and m1' :: "'id \<Rightarrow> 'val" assumes "d_Bisimulation_Up_To_USdB d R"   and "i < length V"   and "(V, V') \<in> R"   and "\<langle>V ! i,m1\<rangle> \<rightarrow> \<langle>W,m2\<rangle>"   and "m1 =\<^bsub>d\<^esub> m1'" shows "\<exists>W' m2'. \<langle>V' ! i,m1'\<rangle> \<rightarrow> \<langle>W',m2'\<rangle> \<and> (W, W') \<in> R \<union> (\<approx>\<^bsub>d\<^esub>) \<and> m2 =\<^bsub>d\<^esub> m2'"
get_def: fixes f :: "'b \<Rightarrow> ('b, 'c) sp\<^sub>\<nu>" shows "get f = In (Get (\<lambda>a. out (f a)))"
compl_le_swap1: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and y :: "'a"   and x :: "'a" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top"   and "less_eq y (uminus x)" shows "less_eq x (uminus y)"
split_spec_exact_def: fixes x :: "'b set" shows "split_spec_exact x = SPEC (\<lambda>(a, b). x = a \<union> b)"
sum_mset_delta': fixes y :: "'c"   and c :: "'b"   and A :: "'c multiset" shows "(\<Sum>x\<in>#A. if y = x then c else (0::'b)) = c * of_nat (count A y)"
\<sigma>_2: shows "strategy_attracts_via p \<sigma> v0 A W"
LcmI: fixes A :: "'a set"   and b :: "'a" assumes "\<And>a. a \<in> A \<Longrightarrow> a dvd b"   and "\<And>c. (\<And>a. a \<in> A \<Longrightarrow> a dvd c) \<Longrightarrow> b dvd c"   and "normalize b = b" shows "b = Lcm A"
subexpr_typing: fixes e1 :: "exp"   and e0 :: "exp"   and CT :: "nat \<Rightarrow> classDef option"   and C :: "nat" assumes "e1 \<in> subexprs(e0)"   and "CT;Map.empty \<turnstile> e0 : C" shows "\<exists>D. CT;Map.empty \<turnstile> e1 : D"
reachable_s_def: fixes PROB :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set"   and s :: "('c, 'd) fmap" shows "reachable_s PROB s \<equiv> {exec_plan s as |as. as \<in> valid_plans PROB}"
domain_subset_of_D: shows "{x *\<^sub>R i + c |x. a \<le> x \<and> x \<le> b} \<subseteq> D"
elims: fixes x :: "formula"   and y :: "formula list" assumes "get_and_list x = y" obtains l :: "formula list" where "x = Ands l" and "y = l"   | v :: "String.literal" and va :: "trm list" where "x = Pred va__ vaa__" and "y = [Pred va__ vaa__]"   | v :: "String.literal" and va :: "formula" and vb :: "formula" where "x = formula.Let vb__ vab__ vba__" and "y = [formula.Let vb__ vab__ vba__]"   | v :: "trm" and va :: "trm" where "x = formula.Eq vc__ vac__" and "y = [formula.Eq vc__ vac__]"   | v :: "trm" and va :: "trm" where "x = Less vd__ vad__" and "y = [Less vd__ vad__]"   | v :: "trm" and va :: "trm" where "x = LessEq ve__ vae__" and "y = [LessEq ve__ vae__]"   | v :: "formula" where "x = Neg vf__" and "y = [Neg vf__]"   | v :: "formula" and va :: "formula" where "x = Or vg__ vaf__" and "y = [Or vg__ vaf__]"   | v :: "formula" and va :: "formula" where "x = And vh__ vag__" and "y = [And vh__ vag__]"   | v :: "formula" where "x = Exists vi__" and "y = [Exists vi__]"   | v :: "nat" and va :: "agg_type \<times> event_data" and vb :: "nat" and vc :: "trm" and vd :: "formula" where "x = Agg vj__ vah__ vbb__ vc vd" and "y = [Agg vj__ vah__ vbb__ vc vd]"   | v :: "\<I>" and va :: "formula" where "x = Prev vk__ vai__" and "y = [Prev vk__ vai__]"   | v :: "\<I>" and va :: "formula" where "x = Next vl__ vaj__" and "y = [Next vl__ vaj__]"   | v :: "formula" and va :: "\<I>" and vb :: "formula" where "x = Since vm__ vak__ vbc__" and "y = [Since vm__ vak__ vbc__]"   | v :: "formula" and va :: "\<I>" and vb :: "formula" where "x = Until vn__ val__ vb" and "y = [Until vn__ val__ vb]"   | v :: "\<I>" and va :: "formula Regex.regex" where "x = MatchF vo__ vam__" and "y = [MatchF vo__ vam__]"   | v :: "\<I>" and va :: "formula Regex.regex" where "x = MatchP v va" and "y = [MatchP v va]"
map_id: fixes t :: "'e T\<^sub>h" shows "map_T\<^sub>h id t = t"
integrable_continuous_real: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a" assumes "continuous_on {a..b} f" shows "f integrable_on {a..b}"
wf_Int_Times: fixes A :: "'d set"   and B :: "'d set" assumes "A \<inter> B = {}" shows "wf (A \<times> B)"
higherPowers\<^sub>f_def: shows "higherPowers\<^sub>f \<equiv> map_fun id (map_fun id (map_fun fmlookup Abs_fmap)) (\<lambda>b i f x. f (inc_above b i x))"
elims: fixes x :: "'b list"   and y :: "'b \<Rightarrow> 'b" assumes "cycle_of_list x = y" obtains i :: "'b" and j :: "'b" and cs :: "'b list" where "x = i # j # cs" and "y = Fun.swap i j id \<circ> cycle_of_list (j # cs)"   | "x = []" and "y = id"   | v :: "'b" where "x = [v]" and "y = id"
prime_factorization_nat_1: fixes n :: "nat" shows "Ball (set (prime_factorization_nat n)) prime"
filtermap_fun_inverse: fixes g :: "'c \<Rightarrow> 'd"   and F :: "'d filter"   and G :: "'c filter"   and f :: "'d \<Rightarrow> 'c" assumes "filterlim g F G"   and "filterlim f G F"   and "\<forall>\<^sub>F x in G. f (g x) = x" shows "filtermap f F = G"
simps_7: fixes y :: "nat" shows "pratt2_max [y]\<^sub>2 \<infinity>\<^sub>2 = \<infinity>\<^sub>2"
div_lt_mult: fixes i :: "'b word"   and k :: "'b word"   and x :: "'b word" assumes "i < k div x"   and "0 < x" shows "i * x < k"
no_paid: fixes s :: "'a list"   and "is" :: "'is"   and q :: "'a" shows "\<forall>((free, paid), uu_)\<in>set_pmf (snd A (s, is) q). paid = []"
cases: fixes a :: "'a"   and P :: "'a set set"   and I :: "'a set" assumes "a \<in> gen P I" obtains i :: "'a" where "a = i" and "i \<in> I"   | J :: "'a set" and j0 :: "'a" and j :: "'a" where "a = j" and "J \<in> P" and "j0 \<in> J" and "j0 \<in> gen P I" and "j \<in> J"
simps_4: fixes dt :: "loc \<Rightarrow> ty option" shows "typeof dt Null = Some NT"
neutr_eq_iff: fixes a :: "'a upred"   and b :: "'a upred" shows "(true = (a \<and> b)) = (a = true \<and> b = true)"
IH: fixes f :: "'a \<Rightarrow> 'b" assumes "f \<in> A \<rightarrow>\<^sub>M B"   and "set xs \<subseteq> space C"   and "\<And>c. c \<in> set xs \<Longrightarrow> (\<lambda>a. case a of (a, b) \<Rightarrow> g a b c) \<in> A \<Otimes>\<^sub>M B \<rightarrow>\<^sub>M B" shows "(\<lambda>x. foldl (g x) (f x) xs) \<in> A \<rightarrow>\<^sub>M B"
simps_2: fixes f :: "dB"   and x :: "dB" shows "norm_pn f (Pure x) = Pure (f \<degree> x)"
C: shows "C = four_block_mat (1\<^sub>m 1) (0\<^sub>m 1 (m - 1)) (0\<^sub>m (m - 1) 1) D"
0: fixes m :: "nat" assumes "0 < m" shows "y m = inv\<^bsub>residue_ring (p ^ m)\<^esub> x m"
induct: fixes x1 :: "'msg"   and x2 :: "'msg"   and P :: "'msg \<Rightarrow> 'msg \<Rightarrow> bool" assumes "hb x1 x2"   and "\<And>m1 i m2. Broadcast m1 \<sqsubset>\<^sup>i Broadcast m2 \<Longrightarrow> P m1 m2"   and "\<And>m1 i m2. Deliver m1 \<sqsubset>\<^sup>i Broadcast m2 \<Longrightarrow> P m1 m2"   and "\<And>m1 m2 m3. \<lbrakk>hb m1 m2; P m1 m2; hb m2 m3; P m2 m3\<rbrakk> \<Longrightarrow> P m1 m3" shows "P x1 x2"
canonical_proj_ext_tl: fixes I :: "nat \<Rightarrow> 'a set"   and n :: "nat" shows "canonical_proj_ext I (Suc n) = (\<lambda>a. (I 0 +> a) # canonical_proj_ext (\<lambda>i. I (Suc i)) n a)"
div_eq_0_conv': fixes m :: "nat"   and n :: "nat" assumes "0 < m" shows "(n div m = 0) = (n < m)"
Id_compose: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set" assumes "f \<in> A \<rightarrow> B"   and "f \<in> extensional A" shows "compose A (\<lambda>y\<in>B. y) f = f"
mono_def: fixes f :: "'b \<Rightarrow> 'b" shows "preord_class.mono f = (\<forall>x y. x \<sqsubseteq> y \<longrightarrow> f x \<sqsubseteq> f y)"
fpxs_val_add_ge: fixes f :: "'b fpxs"   and g :: "'b fpxs" assumes "f + g \<noteq> 0" shows "min (fpxs_val f) (fpxs_val g) \<le> fpxs_val (f + g)"
pcr_cr_eq: shows "pcr_linear_poly = cr_linear_poly"
discI: fixes option :: "'g option"   and x2 :: "'g" assumes "option = Some x2" shows "option \<noteq> None"
ainvf_bijec: fixes F :: "('c, 'd) aGroup_scheme"   and G :: "('e, 'f) aGroup_scheme"   and f :: "'c \<Rightarrow> 'e" assumes "aGroup F"   and "aGroup G"   and "bijec\<^bsub>F,G\<^esub> f" shows "bijec\<^bsub>G,F\<^esub> (ainvf\<^bsub>F,G\<^esub> f)"
iteration_step_measure: fixes ES :: "('b list set \<times> 'b trm set) set"   and X :: "'b list set"   and xrhs :: "'b trm set" assumes "invariant ES"   and "(X, xrhs) \<in> ES"   and "Cond ES" shows "(Iter X ES, ES) \<in> measure card"
norci_def: fixes e1 :: "nat"   and e2 :: "nat"   and s :: "bddi" shows "norci e1 e2 s \<equiv> orci e1 e2 s \<bind> (\<lambda>(r, s). notci r s)"
collapse_2: fixes literal :: "'b literal" assumes "is_neg literal" shows "Neg (atm_of literal) = literal"
min_less_iff_disj: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and y :: "'g"   and z :: "'g" assumes "class.linorder less_eq less" shows "less (ord.min less_eq x y) z = (less x z \<or> less y z)"
run_catch_env: fixes catch :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and m1 :: "('b, 'a) envT"   and m2 :: "('b, 'a) envT"   and r :: "'b" shows "run_env (catch_env catch m1 m2) r = catch (run_env m1 r) (run_env m2 r)"
setdist_le_dist: fixes x :: "'a"   and s :: "'a set"   and y :: "'a"   and t :: "'a set" assumes "x \<in> s"   and "y \<in> t" shows "setdist s t \<le> dist x y"
desargues_config_not_col_5: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and M :: "Points"   and N :: "Points"   and P :: "Points"   and R :: "Points" assumes "desargues_config A B C A' B' C' M N P R" shows "\<not> col B B' A'"
simps_2: fixes \<theta> :: "(name \<times> trm) list"   and e\<^sub>1 :: "trm"   and e\<^sub>2 :: "trm" shows "\<theta><App e\<^sub>1 e\<^sub>2> = App (\<theta><e\<^sub>1>) (\<theta><e\<^sub>2>)"
m_closed: fixes G :: "('f, 'g) monoid_scheme"   and x :: "'f"   and y :: "'f" assumes "Group.monoid G"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> y \<in> carrier G"
indRelTEQ_impl_TRel_is_strong_reduction_bisimulation: fixes TRel :: "('procT \<times> 'procT) set" assumes "(\<forall>P Q P'. P \<sim>\<lbrakk>\<cdot>\<rbrakk>T<TRel> Q \<and> P \<longmapsto>STCal Source Target P' \<longrightarrow> (\<exists>Q'. Q \<longmapsto>STCal Source Target Q' \<and> P' \<sim>\<lbrakk>\<cdot>\<rbrakk>T<TRel> Q')) \<and> (\<forall>P Q Q'. P \<sim>\<lbrakk>\<cdot>\<rbrakk>T<TRel> Q \<and> Q \<longmapsto>STCal Source Target Q' \<longrightarrow> (\<exists>P'. P \<longmapsto>STCal Source Target P' \<and> P' \<sim>\<lbrakk>\<cdot>\<rbrakk>T<TRel> Q'))" shows "strong_reduction_simulation (TRel\<^sup>*) Target \<and> (\<forall>P Q Q'. (P, Q) \<in> TRel\<^sup>* \<and> Q \<longmapsto>Target Q' \<longrightarrow> (\<exists>P'. P \<longmapsto>Target P' \<and> (P', Q') \<in> TRel\<^sup>*))"
zero_vec_def: fixes n :: "nat" shows "0\<^sub>v n \<equiv> vec n (\<lambda>i. 0::'a)"
compl_inf_bot: fixes x :: "'b upred" shows "(\<not> x \<and> x) = false"
simps_5: fixes f1 :: "qtname \<Rightarrow> 'a"   and f2 :: "ty \<Rightarrow> int \<Rightarrow> 'a"   and x1 :: "qtname" shows "(case CInst x1 of CInst x \<Rightarrow> f1 x | Arr x xa \<Rightarrow> f2 x xa) = f1 x1"
cast\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>2\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_some: fixes document :: "('m, 'n, 'o, (\<lparr>RShadowRoot.nothing :: unit, mode :: shadow_root_mode, child_nodes :: (_) node_ptr list, \<dots> :: 'p option\<rparr> + 'q) option) RDocument_scheme"   and shadow_root :: "('m, 'n, 'o, 'p option) RShadowRoot_scheme" shows "(cast document = Some shadow_root) = (cast shadow_root = document)"
le_comp_left_right: fixes x :: "'a"   and y :: "'a"   and u :: "'a"   and v :: "'a" assumes "x \<le> y"   and "u \<le> v" shows "x * u \<le> y * v"
mult_le_cancel_iff2: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "(0::'a) < z" shows "(z * x \<le> z * y) = (x \<le> y)"
composite_simps_4: shows "trg ((\<nu> \<star> \<pi>) \<star> \<rho>) = trg \<nu>"
not_accp_down: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "\<not> Wellfounded.accp R x" obtains z :: "'c" where "R z x" and "\<not> Wellfounded.accp R z"
S_def: shows "S = {w \<in> space M. stake 0 w = stake 0 x}"
that: fixes \<alpha>s :: "V list" assumes "list.set \<alpha>s \<subseteq> ON"   and "sorted_wrt (\<lambda>x y. y \<le> x) \<alpha>s"   and "\<alpha> = \<omega>_sum \<alpha>s" shows "thesis"
pdigraph_baseI: fixes \<alpha> :: "V"   and I :: "V"   and \<AA> :: "V \<Rightarrow> V" assumes "\<Z> \<alpha>"   and "\<And>i. i \<in>\<^sub>\<circ> I \<Longrightarrow> digraph \<alpha> (\<AA> i)"   and "I \<in>\<^sub>\<circ> Vset \<alpha>" shows "pdigraph_base \<alpha> I \<AA>"
bheight_paint_Red: fixes t :: "('b \<times> color) tree" assumes "color t = Black" shows "bheight (paint Red t) = bheight t - 1"
M_elem_map_eq: fixes M2 :: "'a set" assumes "M2 \<in> M" shows "M1 #> (SOME x. x \<in> carrier G \<and> M1 #> x = M2) = M2"
assms_3: shows "\<forall>\<^sub>F x in F. 0 \<le> f x"
rtranclp_mult_poly_p_mult_ideal: fixes q :: "(char list multiset \<times> int) multiset"   and p :: "(char list multiset \<times> int) multiset"   and r :: "(char list multiset \<times> int) multiset"   and p' :: "(char list multiset \<times> int) multiset"   and r' :: "(char list multiset \<times> int) multiset" assumes "(mult_poly_p q)\<^sup>*\<^sup>* (p, r) (p', r')" shows "polynomial_of_mset p' * polynomial_of_mset q + polynomial_of_mset r' - (polynomial_of_mset p * polynomial_of_mset q + polynomial_of_mset r) \<in> More_Modules.ideal polynomial_bool"
\<J>_0_0_subset_Gen: shows "\<J> 0 0 \<subseteq> Gen (\<P> 0 0)"
cring_simprules_1: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a" assumes "cring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> y \<in> carrier R"
gWlsBinp_def: fixes MOD :: "('h, 'i, 'varSort, 'sort, 'j, 'k, 'l, 'm, 'n) Recursion.model_scheme"   and delta :: "'opSym"   and binp :: "'bindex \<Rightarrow> 'm option" shows "gWlsBinp MOD delta binp \<equiv> wlsOpS delta \<and> sameDom (barOf delta) binp \<and> liftAll2 (gWlsAbs MOD) (barOf delta) binp"
open_empty: fixes "open" :: "'a set \<Rightarrow> bool" assumes "class.topological_space open" shows "open {}"
all_args_def: fixes f :: "'at \<Rightarrow> ('aq, 'ar, 'as) prog" shows "all_args f = (\<Union>arg. {f arg})"
assms: shows "p \<in> dgrad_sig_set' (length fs) dgrad"
r_e2stack: fixes e :: "nat" shows "eval r_e2stack [e] \<down>= e2stack e"
arrow_of_spans_axioms: shows "arrow_of_spans (\<cdot>) (mkIde f0 f1)"
fun_rel_comp_dist: fixes R1 :: "('g \<times> 'k) set"   and R2 :: "('h \<times> 'l) set"   and R3 :: "('k \<times> 'i) set"   and R4 :: "('l \<times> 'j) set" shows "(R1 \<rightarrow> R2) O (R3 \<rightarrow> R4) \<subseteq> R1 O R3 \<rightarrow> R2 O R4"
All2_SUCC_I: fixes i :: "name"   and t :: "tm"   and H :: "fm set"   and A :: "fm" assumes "atom i \<sharp> t"   and "H \<turnstile> All2 i t A"   and "H \<turnstile> A(i::=t)" shows "H \<turnstile> All2 i (SUCC t) A"
card_dc_crypto: shows "card dc_crypto = n * 2 ^ n"
leading_coeff_0_iff: fixes p :: "'b poly" shows "(lead_coeff p = (0::'b)) = (p = 0)"
eq_iff: fixes x :: "'a list"   and y :: "'a list" shows "(x = y) = (prefix x y \<and> prefix y x)"
vsubset_vintersection_rightD_2: fixes A :: "V"   and B :: "V"   and C :: "V" assumes "A \<subseteq>\<^sub>\<circ> B \<inter>\<^sub>\<circ> C" shows "A \<subseteq>\<^sub>\<circ> C"
poly_mult_rat_irreducible: fixes p :: "int poly"   and r :: "rat" assumes "irreducible p"   and "degree p \<noteq> 0"   and "r \<noteq> 0" shows "irreducible (cf_pos_poly (poly_mult_rat r p))"
prod_list_diag_prod: fixes A :: "'b mat" shows "prod_list (diag_mat A) = (\<Prod>i = 0..<dim_row A. A $$ (i, i))"
sorted_remdups: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq xs" shows "linorder.sorted less_eq (remdups xs)"
sup_inf_distrib2: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and y :: "'b"   and z :: "'b"   and x :: "'b" assumes "class.distrib_lattice inf less_eq less sup" shows "sup (inf y z) x = inf (sup y x) (sup z x)"
Pi_pmf_return_pmf': fixes A :: "'b set"   and dflt :: "'c" assumes "finite A" shows "Pi_pmf A dflt (\<lambda>_. return_pmf dflt) = return_pmf (\<lambda>_. dflt)"
top_add_to_counter: fixes env :: "v sem_env"   and s :: "'b state"   and d :: "top0"   and s' :: "'b state"   and r :: "(v sem_env, v) result"   and extra :: "nat" assumes "evaluate_top True env s d (s', r)"   and "r \<noteq> Rerr (Rabort Rtimeout_error)" shows "evaluate_top True env (update_clock (\<lambda>_. clock s + extra) s) d (update_clock (\<lambda>_. clock s' + extra) s', r)"
min_product_def: fixes A :: "'b fset set"   and B :: "'b fset set" shows "A \<otimes>\<^sub>m B = min_set (A \<otimes> B)"
simps_4: fixes p :: "('d, 'e, 'f) formula" shows "SIGP (? p) = SIGF p"
VDM_valid_def: fixes c :: "OBJ"   and A :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool" shows "( \<Turnstile> c : A ) = (\<forall>s t. s , c \<Down> t  \<longrightarrow> A s t)"
mono_filter_less': fixes a1 :: "'a"   and b1 :: "'a"   and a2 :: "'a"   and b2 :: "'a"   and bv :: "bool" assumes "a1 \<sqsubseteq> b1"   and "a2 \<sqsubseteq> b2" shows "filter_less' bv a1 a2 \<sqsubseteq> filter_less' bv b1 b2"
foldr_True_set: fixes f :: "'c \<Rightarrow> bool"   and l :: "'c list" shows "foldr (\<lambda>x. (\<and>) (f x)) l True = (\<forall>x\<in>set l. f x)"
that: fixes \<gamma> :: "'a \<Rightarrow> 'a set" assumes "gauge \<gamma>"   and "\<And>c i S h. \<lbrakk>c \<in> cbox a b; i \<in> Basis; S tagged_partial_division_of cbox a b; \<gamma> fine S; h \<in> F; \<And>x K. (x, K) \<in> S \<Longrightarrow> K \<inter> {x. x \<bullet> i = c \<bullet> i} \<noteq> {}\<rbrakk> \<Longrightarrow> (\<Sum>(x, K)\<in>S. norm (integral K h)) < \<epsilon>" shows "thesis"
asInp_def: fixes qinp :: "'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qTerm option" shows "asInp qinp \<equiv> lift asTerm qinp"
closed_Int: fixes "open" :: "'b set \<Rightarrow> bool"   and S :: "'b set"   and T :: "'b set" assumes "class.topological_space open"   and "topological_space.closed open S"   and "topological_space.closed open T" shows "topological_space.closed open (S \<inter> T)"
martingale_def: fixes M :: "'c measure"   and F :: "'d \<Rightarrow> 'c measure"   and X :: "'d \<Rightarrow> 'c \<Rightarrow> real" shows "martingale M F X = (Filtration.filtration M F \<and> (\<forall>t. integrable M (X t)) \<and> borel_adapt_stoch_proc F X \<and> (\<forall>t s. t \<le> s \<longrightarrow> (AE w in M. real_cond_exp M (F t) (X s) w = X t w)))"
assms_3: shows "dist p c \<le> dist v (p $ k)"
always_imp_mono_1: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'b"   and P' :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" assumes "(\<box>P) \<sigma>"   and "(\<box>(\<lambda>s. P s \<longrightarrow> P' s)) \<sigma>" shows "(\<box>P') \<sigma>"
assms: fixes k :: "nat"   and n :: "nat" assumes "\<sigma> (Suc k) = \<sigma> k"   and "k < n" shows "\<sigma> n = \<sigma> k"
coeff_Poly_list_of_vec_nth: fixes x :: "nat"   and A :: "'b mat"   and i :: "nat" assumes "x < dim_col A" shows "coeff (Poly (list_of_vec (row A i))) x = A $$ (i, x)"
sum_is_minimal: fixes N1 :: "'c set"   and N2 :: "'c set"   and N :: "'c set" assumes "LinearCombinations.submodule K N1 V"   and "LinearCombinations.submodule K N2 V"   and "LinearCombinations.submodule K N V" shows "(subspace_sum N1 N2 \<subseteq> N) = (N1 \<subseteq> N \<and> N2 \<subseteq> N)"
eq_OO: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "(=) OO R = R"
f': assumes "x \<in> S \<union> closure S \<inter> closure T" shows "(f has_derivative f' x) (at x within S \<union> closure S \<inter> closure T)"
snapshot_subset: fixes PROB :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) set"   and s :: "('a, 'b) fmap" shows "snapshot PROB s \<subseteq> PROB"
that_2: fixes c :: "'a"   and e :: "real"   and xs' :: "('a \<times> real) msllist" assumes "xs = MSLCons (c, e) xs'" shows "thesis"
app_exec: fixes cs :: "com list"   and s :: "state"   and cs' :: "com list"   and s' :: "state"   and cs2 :: "com list" assumes "(cs, s) \<rightarrow> (cs', s')" shows "(cs @ cs2, s) \<rightarrow> (cs' @ cs2, s')"
is_msg_defs_1: fixes \<xi> :: "'a state_scheme" shows "is_pkt \<xi> \<equiv> case msg \<xi> of Pkt num' sid' \<Rightarrow> {\<xi>\<lparr>num := num', sid := sid'\<rparr>} | Newpkt nat1 nat2 \<Rightarrow> {}"
coprime_1: fixes c1 :: "complex poly"   and c2 :: "complex poly"   and m :: "nat"   and n :: "nat" assumes "\<not> coprime c1 c2"   and "(c1, m) \<in> set cs"   and "(c2, n) \<in> set cs" shows "c1 = c2"
left_option_is_option: fixes x :: "game"   and g :: "game" assumes "zin x (left_options g)" shows "zin x (options g)"
is_group: shows "Group.group (DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I)"
cSUP_upper: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'c"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'b" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "x \<in> A"   and "preorder.bdd_above less_eq (f ` A)" shows "less_eq (f x) (Sup (f ` A))"