hcju/isabelleps huggingface dataset:
bind_returns_result_E: fixes h :: "'e"   and f :: "('e, 'f, 'h) prog"   and g :: "'h \<Rightarrow> ('e, 'f, 'g) prog"   and y :: "'g" assumes "h \<turnstile> f \<bind> g \<rightarrow>\<^sub>r y" obtains x :: "'h" and h' :: "'e" where "h \<turnstile> f \<rightarrow>\<^sub>r x" and "h \<turnstile> f \<rightarrow>\<^sub>h h'" and "h' \<turnstile> g x \<rightarrow>\<^sub>r y"
continuous_openin_preimage_gen: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and T :: "'d set" assumes "continuous_on S f"   and "open T" shows "openin (top_of_set S) (S \<inter> f -` T)"
nth_sequence_of_inv_into: fixes enum :: "nat \<Rightarrow> 'b"   and A :: "'b set"   and a :: "'b"   and f :: "'b \<Rightarrow> 'c" assumes "bij_betw enum {0..<card A} A"   and "a \<in> A" shows "sequence_of A enum f ! inv_into {0..<card A} enum a = f a"
chamber_retraction1: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and C :: "'b set" assumes "ChamberComplexRetraction X f"   and "C \<in> f \<turnstile> ChamberComplex.chamber_system X" shows "fixespointwise f C"
bot_scene_def: shows "\<bottom>\<^sub>S \<equiv> Abs_scene (\<lambda>x y. x)"
2: shows "1 - z\<^sup>2 \<notin> \<real>\<^sub>\<le>\<^sub>0"
strict_mono_on_imp_mono_on: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "strict_mono_on f A" shows "mono_on f A"
squarefreeD: fixes n :: "'a"   and x :: "'a" assumes "squarefree n"   and "x\<^sup>2 dvd n" shows "x dvd (1::'a)"
not_produces_imp_notin_production: fixes N :: "'b literal multiset set"   and C :: "'b literal multiset"   and A :: "'b" assumes "production N C \<noteq> {A}" shows "A \<notin> production N C"
triv_normal_subgroup: shows "{\<one>} \<lhd> G"
hom_restrict: fixes G :: "('c, 'd) monoid_scheme"   and h :: "'c \<Rightarrow> 'e"   and H :: "('e, 'f) monoid_scheme"   and t :: "'c \<Rightarrow> 'e" assumes "Group.group G"   and "h \<in> hom G H"   and "\<And>g. g \<in> carrier G \<Longrightarrow> h g = t g" shows "t \<in> hom G H"
fresh_star_def: fixes as :: "atom set"   and x :: "'e" shows "as \<sharp>* x \<equiv> \<forall>a\<in>as. a \<sharp> x"
hom_closed: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set" assumes "x \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))" shows "hom_induced (int n) (nsphere n) (equator n) (nsphere n) (equator n) f x \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))"
hyps: fixes rs :: "'r list" assumes "set rs \<subseteq> R"   and "set ms \<subseteq> M" shows "rs \<bullet>\<cdot> ms \<in> M"
iNext_False: fixes t0 :: "nat"   and I :: "nat set" shows "\<not> (\<circle> t t0 I. False)"
simps_3: fixes \<Gamma> :: "(name \<times> ty) list"   and s :: "trm"   and t :: "trm"   and T\<^sub>1 :: "ty"   and T\<^sub>2 :: "ty" shows "\<Gamma> \<turnstile> s is t : T\<^sub>1\<rightarrow>T\<^sub>2 = (\<forall>\<Gamma>'' s' t'.  \<Gamma> \<subseteq> \<Gamma>''  \<longrightarrow> valid \<Gamma>'' \<longrightarrow> \<Gamma>'' \<turnstile> s' is t' : T\<^sub>1 \<longrightarrow> \<Gamma>'' \<turnstile> App s s' is App t t' : T\<^sub>2)"
seqs_ne: fixes K :: "nat set"   and j0 :: "nat"   and j :: "nat" assumes "K \<in> \<K> j0 j"   and "j \<le> j0" shows "seqs j0 j K \<in> lists (- {[]})"
BseqI: fixes K :: "real"   and X :: "nat \<Rightarrow> 'a" assumes "0 < K"   and "\<forall>n. norm (X n) \<le> K" shows "Bseq X"
count_derangements_approximation: fixes n :: "nat" assumes "n \<noteq> 0" shows "\<bar>real (count_derangements n) - fact n / exp 1\<bar> < 1 / 2"
card_of_cong: fixes r :: "('d \<times> 'd) set"   and r' :: "('e \<times> 'e) set" assumes "r =o r'" shows "|Field r| =o |Field r'|"
linear_ow_def: fixes U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d set"   and plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd"   and scale\<^sub>1 :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f set"   and plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f"   and scale\<^sub>2 :: "'e \<Rightarrow> 'f \<Rightarrow> 'f"   and f :: "'d \<Rightarrow> 'f" shows "linear_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 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 f \<equiv> 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 \<and> 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 \<and> module_hom_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 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 f"
g_def: shows "g = (\<lambda>i. bnd ((upd ^^ i) init))"
jump_ok_GotoD: fixes ins :: "'a instr list"   and n :: "nat"   and n' :: "nat"   and pc :: "nat"   and m :: "int" assumes "jump_ok ins n n'"   and "ins ! pc = Goto m"   and "pc < length ins" shows "- int (pc + n) \<le> m \<and> m < int (length ins - pc + n')"
countable_basis: fixes p :: "'d set \<Rightarrow> bool"   and B :: "'d set set" assumes "countable_basis p B" shows "countable B"
altts_def: shows "altts = alternating_list (Suc (S_length w)) t s"
assms_2: shows "\<forall>i. \<pi> i \<noteq> local.return"
trm_in_hom: fixes a :: "'a" assumes "ide a" shows "\<guillemotleft>\<t>[a] : a \<rightarrow> \<I>\<guillemotright>"
find_base_vector_def: fixes A :: "'c mat" shows "find_base_vector A \<equiv> let pp = pivot_positions A; cands = filter (\<lambda>j. j \<notin> set (map snd pp)) [0..<dim_col A] in non_pivot_base A pp (hd cands)"
mod_ring_gen_axioms: shows "mod_ring_gen ff_ops R p"
phl_export2: fixes z :: "'a"   and w :: "'a"   and x :: "'a"   and y :: "'a" assumes "\<lparr>z\<rparr>w\<lparr>w \<cdot> z \<cdot> w\<rparr>"   and "\<lparr>x\<rparr>y\<lparr>y \<cdot> z\<rparr>" shows "\<lparr>x\<rparr>y \<cdot> w\<lparr>y \<cdot> w \<cdot> (z \<cdot> w)\<rparr>"
fmrestrict_set_drop: fixes S :: "'c set"   and b :: "'c"   and m :: "('c, 'd) fmap" shows "fmrestrict_set S (fmdrop b m) = fmrestrict_set (S - {b}) m"
assms: shows "[M]\<^bsub>v,n\<^esub> \<noteq> {}"
assms_2: fixes X :: "'a set" assumes "X \<subseteq> C" shows "f X \<subseteq> C"
tensor_arr_unity: fixes f :: "'c local.term set" assumes "arr f" shows "f \<otimes>\<^sub>S \<I> = f"
limit_def: fixes x :: "nat \<Rightarrow> 'b" shows "limit x \<equiv> {a. \<exists>\<^sub>\<infinity>n. x n = a}"
heterogenous_same_type_cblinfun_def: fixes x :: "'b itself"   and y :: "'c itself" shows "heterogenous_same_type_cblinfun x y = (unitary heterogenous_cblinfun_id \<and> unitary heterogenous_cblinfun_id)"
root_main'_newton_pos: fixes p :: "nat"   and pm :: "nat"   and x :: "int"   and n :: "int"   and y :: "int"   and b :: "bool" assumes "fixed_root p pm"   and "0 \<le> x"   and "0 \<le> n" shows "root_int_main' pm (int pm) (int p) x n = fixed_root.root_newton_int_main p pm x n \<and> (root_int_main' pm (int pm) (int p) x n = (y, b) \<longrightarrow> 0 \<le> y \<and> y ^ p \<le> n \<and> b = (y ^ p = n))"
less_divide_eq_numeral1_1: fixes a :: "'b"   and b :: "'b"   and w :: "num" shows "(a < b / numeral w) = (a * numeral w < b)"
succ_def: fixes G :: "'b graph"   and v :: "'b" shows "succ G v \<equiv> {v'. (v, v') \<in> edges G}"
refl: fixes x :: "'b" assumes "x \<in> carrier" shows "R j x x"
subset_eq: fixes A :: "'b set"   and B :: "'b set" shows "((A::'b set) \<subseteq> (B::'b set)) = (\<forall>x::'b\<in>A. x \<in> B)"
simps_1: fixes x :: "brick list"   and y :: "brick list" shows "basic x \<otimes> basic y = basic (x \<otimes> y)"
in_list_Suc_iff: fixes xs :: "'b list"   and n :: "nat"   and A :: "'b set" shows "(xs \<in> list (Suc n) A) = (\<exists>y\<in>A. \<exists>ys\<in>list n A. xs = y # ys)"
setdist_sym: fixes S :: "'c set"   and T :: "'c set" shows "setdist S T = setdist T S"
std\<^sub>1_renames: fixes C\<^sub>1 :: "fterm literal set" shows "var_renaming_of C\<^sub>1 (C\<^sub>1 \<cdot>\<^sub>l\<^sub>s (\<lambda>x. \<epsilon> (''1'' @ x)))"
simps_3: fixes t :: "'a tree" shows "right (mirror t) = mirror (left t)"
triangle_not_betw_abc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "\<triangle> a b c" shows "\<not> [[a b c]]"
split_beta': fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" shows "(\<lambda>(x, y). f x y) = (\<lambda>x. f (fst x) (snd x))"
less_linear: fixes x :: "'a"   and y :: "'a" shows "x \<sqsubset> y \<or> x = y \<or> y \<sqsubset> x"
fps_divide_unit_factor_denom: fixes g :: "'a fps"   and f :: "'a fps" assumes "subdegree g \<le> subdegree f" shows "f / unit_factor g = Abs_fps (\<lambda>k. if k < subdegree g then 0::'a else fps_nth (f / g) (k - subdegree g))"
when_def: fixes zero :: "'a"   and a :: "'a"   and P :: "bool" shows "zero.when zero a P = (if P then a else zero)"
run_one_step_trap_result: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and ves :: "v list"   and s' :: "s"   and vs' :: "v list"   and res :: "res_step" assumes "run_one_step d i (s, vs, ves, Trap) = (s', vs', res)" shows "\<exists>e. res = RSCrash e"
nested_swap': fixes a :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and n :: "nat" shows "(\<Sum>i\<le>n. sum (a i) {..<i}) = (\<Sum>j<n. \<Sum>i = Suc j..n. a i j)"
rt_arg: fixes u :: "'a"   and u' :: "'a"   and t :: "'a" assumes "R\<^sup>*\<^sup>* u u'" shows "R\<^sup>*\<^sup>* (app t u) (app t u')"
interval_subset_is_interval: fixes S :: "'b set"   and a :: "'b"   and b :: "'b" assumes "is_interval S" shows "(cbox a b \<subseteq> S) = (cbox a b = {} \<or> a \<in> S \<and> b \<in> S)"
vec_to_iarray_inner: fixes a :: "(real, 'b) vec"   and b :: "(real, 'b) vec" shows "a \<bullet> b = vec_to_iarray a \<bullet>i vec_to_iarray b"
integer_group_def: shows "integer_group = \<lparr>carrier = UNIV, monoid.mult = (+), one = 0\<rparr>"
BIT: shows "T\<^sub>p_on_rand' BIT s qs = 75 / 10\<^sup>2 * real (length qs) - 5 / 10"
prems_1: shows "braun \<langle>t1_, x2_, t2_\<rangle>"
components_isotone: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "components x \<le> components y"
longest_common_prefix_max_prefix: fixes ps :: "'b list"   and xs :: "'b list"   and ys :: "'b list" assumes "ps \<le>p xs"   and "ps \<le>p ys" shows "ps \<le>p xs \<and>\<^sub>p ys"
bunch_of_lemmata_about_matches_4: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and m :: "'b match_expr"   and a :: "action"   and p :: "'c" shows "matches \<gamma> (MatchNot (MatchNot m)) a p = matches \<gamma> m a p"
trans: fixes b :: "'b list"   and a :: "'b list"   and c :: "'b list" assumes "b \<le>s a"   and "c \<le>s b" shows "c \<le>s a"
convex_PInfty_outside: fixes f :: "'b \<Rightarrow> ereal"   and S :: "'b set" assumes "Lower_Semicontinuous.convex_on UNIV f"   and "convex S" shows "Lower_Semicontinuous.convex_on UNIV (\<lambda>x. if x \<in> S then f x else \<infinity>)"
list_slice2_Nil: fixes k :: "nat" shows "list_slice2 [] k = []"
scale_minus_left: fixes a :: "'a"   and x :: "'b" shows "- a *s x = - (a *s x)"
cone_closed_uminus: fixes a :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and hU :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set" assumes "a \<in> cone hU" shows "- a \<in> cone hU"
nat_pow_comm: fixes x :: "'a"   and n :: "nat"   and m :: "nat" assumes "x \<in> carrier R" shows "x [^] n \<otimes> x [^] m = x [^] m \<otimes> x [^] n"
is_app_def: fixes size :: "'a \<Rightarrow> nat"   and frees :: "'a \<Rightarrow> name fset"   and subst :: "'a \<Rightarrow> (name, 'a) fmap \<Rightarrow> 'a"   and "consts" :: "'a \<Rightarrow> name fset"   and app :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unapp :: "'a \<Rightarrow> ('a \<times> 'a) option"   and const :: "name \<Rightarrow> 'a"   and unconst :: "'a \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'a"   and unfree :: "'a \<Rightarrow> name option"   and t :: "'a" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "pre_term.is_app unapp t = (unapp t \<noteq> None)"
length_takeWhile_Not_replicate_False: fixes n :: "nat"   and ls :: "bool list" shows "length (takeWhile Not (replicate n False @ ls)) = n + length (takeWhile Not ls)"
search_enum_nonempty: fixes b :: "nat"   and i :: "nat"   and y\<^sub>0 :: "nat"   and e :: "nat"   and x :: "nat" assumes "s b \<down>= i"   and "\<phi> i y\<^sub>0 \<down>"   and "e = the (eval r_enumdom [i, x])" shows "search_enum x b \<down>= (if e_length b \<le> e then 0 else 1)"
tf: fixes S :: "'a set" assumes "S \<in> \<F>" shows "open (tf S) \<and> S = U \<inter> tf S"
is_bezout_ext_euclid_ext2: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Gcd :: "'b set \<Rightarrow> 'b"   and Lcm :: "'b set \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b" assumes "class.euclidean_ring_gcd divide plus minus zero times one modulo euclidean_size unit_factor normalize gcd lcm Gcd Lcm uminus" shows "bezout_ring.is_bezout_ext times plus uminus (euclidean_ring_gcd.euclid_ext2 divide minus zero times one modulo unit_factor normalize gcd uminus)"
invar_butlast: fixes bq :: "('c, 'd) SkewBinomialTree list"   and t :: "('c, 'd) SkewBinomialTree" assumes "invar (bq @ [t])" shows "invar bq"
indRelLTPO_impl_TRel_is_strong_reduction_simulation: fixes TRel :: "('procT \<times> 'procT) set" assumes "\<forall>P Q P'. P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>LT<TRel> Q \<and> P \<longmapsto>STCal Source Target P' \<longrightarrow> (\<exists>Q'. Q \<longmapsto>STCal Source Target Q' \<and> P' \<lesssim>\<lbrakk>\<cdot>\<rbrakk>LT<TRel> Q')" shows "strong_reduction_simulation (TRel\<^sup>+) Target"
iso_char: fixes f :: "'a" shows "hom.iso f = (local.iso f \<and> hom.arr f \<and> hom.arr (local.inv f))"
is_Send_def: fixes stateful_strand_step :: "('e, 'f) stateful_strand_step" shows "stateful_strand_step.is_Send stateful_strand_step = (\<exists>x1. stateful_strand_step = send\<langle>x1\<rangle>)"
table_def: fixes n :: "nat"   and V :: "nat set"   and X :: "'a option list set" shows "table n V X = (\<forall>x\<in>X. wf_tuple n V x)"
add_neg_numeral_special_2: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and one :: "'c"   and m :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "plus (uminus (numeral.numeral one plus m)) one = neg_numeral.sub minus plus one num.One m"
length_filter_disj_or_conv: fixes xs :: "'a list"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "\<And>x. x \<in> set xs \<longrightarrow> \<not> (P x \<and> Q x)" shows "length (filter (\<lambda>x. P x \<or> Q x) xs) = length (filter P xs) + length (filter Q xs)"
conj_zero_right: fixes x :: "'a" shows "x \<^bold>\<sqinter> \<zero> = \<zero>"
combine_inv1_1: fixes lt :: "('a, 'b) rbt"   and rt :: "('a, 'b) rbt" assumes "inv1 lt"   and "inv1 rt"   and "color_of lt = B"   and "color_of rt = B" shows "inv1 (combine lt rt)"
set_spmf_return_pmf: fixes x :: "'a option" shows "set_spmf (return_pmf x) = set_option x"
fps_catalan_def: shows "fps_catalan = Abs_fps (of_nat \<circ> catalan)"
exec: shows "\<lfloor>\<sigma>'\<rfloor> = exec (P, None, h, (stk, loc, C, M, pc, ics) # frs, sh)"
openin_relative_to: fixes X :: "'c topology"   and S :: "'c set" shows "openin X relative_to S = openin (subtopology X S)"
comp_char: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Arr :: "'b \<Rightarrow> bool"   and g :: "'b"   and f :: "'b" assumes "subcategory C Arr" shows "subcategory.comp C Arr g f = (if partial_magma.arr (subcategory.comp C Arr) f \<and> partial_magma.arr (subcategory.comp C Arr) g \<and> partial_magma.arr C (C g f) then C g f else partial_magma.null C)"
1: fixes aa :: "ereal"   and ca :: "ereal"   and ba :: "ereal" assumes "interval_lebesgue_integrable lborel aa ca f"   and "aa \<le> ba"   and "ba \<le> ca" shows "interval_lebesgue_integral lborel aa ba f + interval_lebesgue_integral lborel ba ca f = interval_lebesgue_integral lborel aa ca f"
ldropn_Suc_LCons: fixes n :: "nat"   and x :: "'b"   and xs :: "'b llist" shows "ldropn (Suc n) (x % xs) = ldropn n xs"
sublist_valid_is_valid: fixes as :: "'c list"   and PROB :: "'c set"   and as' :: "'c list" assumes "as \<in> valid_plans PROB"   and "subseq as' as" shows "as' \<in> valid_plans PROB"
step_t_z_dbm: fixes D_inv :: "nat \<Rightarrow> nat \<Rightarrow> 'e DBMEntry"   and A :: "('g \<times> ('f, 'e) cconstraint \<times> 'h \<times> 'f list \<times> 'g) set \<times> ('g \<Rightarrow> ('f, 'e) cconstraint)"   and l :: "'g"   and v :: "'f \<Rightarrow> nat"   and D :: "nat \<Rightarrow> nat \<Rightarrow> 'e DBMEntry"   and n :: "nat" assumes "D_inv = abstr (inv_of A l) (\<lambda>i j. \<infinity>) v" shows "A \<turnstile> \<langle>l, D\<rangle> \<leadsto>\<^bsub>v,n\<^esub> \<langle>l, And (up (And D D_inv)) D_inv\<rangle>"
seqE: fixes g :: "'a"   and f :: "'a" assumes "Adj.C.seq g f" obtains "Adj.C.arr f" and "Adj.C.arr g" and "Adj.C.dom g = Adj.C.cod f"
chain_homotopic_singular_subdivision: obtains h :: "nat \<Rightarrow> (((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int) \<Rightarrow> ((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" where "\<And>p. h p 0 = 0" and "\<And>p c1 c2. h p (c1 - c2) = h p c1 - h p c2" and "\<And>p X c. singular_chain p X c \<Longrightarrow> singular_chain (Suc p) X (h p c)" and "\<And>p X c. singular_chain p X c \<Longrightarrow> chain_boundary (Suc p) (h p c) + h (p - Suc 0) (chain_boundary p c) = singular_subdivision p c - c"
assms_1: shows "l cd\<^bsup>\<pi>\<^esup>\<rightarrow> k"
discrete: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and modulo :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and euclidean_size :: "'d \<Rightarrow> nat"   and division_segment :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and divmod :: "num \<Rightarrow> num \<Rightarrow> 'd \<times> 'd"   and divmod_step :: "num \<Rightarrow> 'd \<times> 'd \<Rightarrow> 'd \<times> 'd"   and a :: "'d"   and b :: "'d" assumes "class.unique_euclidean_semiring_numeral divide plus minus zero times one modulo euclidean_size division_segment less_eq less divmod divmod_step" shows "less a b = less_eq (plus a one) b"
sorted_fst_take_less_hd_drop: fixes ps :: "(int \<times> int) list"   and n :: "nat" assumes "sorted_fst ps"   and "n < length ps" shows "\<forall>p\<in>set (take n ps). fst p \<le> fst (hd (drop n ps))"
wlsBinp_castOp: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set" assumes "isOp s X" shows "wlsBinp (fst3 (castOp s X)) (trd3 (castOp s X))"
gauss_jordan_def: fixes A :: "'c mat"   and B :: "'c mat" shows "gauss_jordan A B \<equiv> gauss_jordan_main A B 0 0"
hyps: assumes "y \<le> x" shows "{\<lambda>nl. nl = xs @ (x - y) # rec_exec (Pr n f g) (xs @ [x - y]) # 0 \<up> (max (n + 3) (max fft gft) - Suc (Suc n)) @ y # anything} [Dec (max (n + 3) (max fft gft)) (length gap + 7)] [+] (gap [+] [Inc n, Dec (Suc n) 3, Goto (Suc 0)]) @ [Dec (Suc (Suc n)) 0, Inc (Suc n), Goto (length gap + 4)] {\<lambda>nl. nl = xs @ x # rec_exec (Pr n f g) (xs @ [x]) # 0 \<up> (max (n + 3) (max fft gft) - Suc n) @ anything}"
finite_dimension_imp_subalgebra: fixes K :: "'a set"   and E :: "'a set" assumes "subfield K R"   and "finite_dimension K E" shows "subalgebra K E R"
assms_3: shows "r\<^sub>1 \<inter> r\<^sub>2 = {}"
hm_it_finish: fixes m :: "'a \<Rightarrow> 'b option"   and p :: "('a, 'b) hashtable"   and m' :: "'a \<Rightarrow> 'b option"   and it :: "nat \<times> ('a \<times> 'b) list \<times> ('a, 'b) hashtable" shows "hm_is_it m p m' it \<Longrightarrow>\<^sub>A is_hashmap m p"
X: fixes i :: "nat" shows "\<exists>\<beta>. (P \<W> Q) (stake i \<sigma> @- \<beta>)"
antisym: fixes x :: "'x \<Rightarrow>\<^sub>0 nat"   and y :: "'x \<Rightarrow>\<^sub>0 nat" assumes "lex_pm x y"   and "lex_pm y x" shows "x = y"
ls_nds_rel_def: fixes R :: "('d \<times> ('e, 'f) node_scheme) set" shows "\<langle>R\<rangle>ls_nds_rel = \<langle>R\<rangle>list_set_rel \<inter> {(uu_, s). inj_on name s}"
effective_matrix_kauff_mat_2: fixes ws :: "wall" assumes "is_tangle_diagram ws" shows "length (kauff_mat ws) = 2 ^ nat (codomain_wall ws)"
fv\<^sub>s\<^sub>s\<^sub>t_Cons: fixes a :: "('d, 'c) stateful_strand_step"   and A :: "('d, 'c) stateful_strand_step list" shows "fv\<^sub>s\<^sub>s\<^sub>t (a # A) = fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p a \<union> fv\<^sub>s\<^sub>s\<^sub>t A"
that: fixes \<sigma>' :: "nat \<Rightarrow> 'g"   and \<zeta>' :: "'l net_state" assumes "((\<sigma>, \<zeta>), \<tau>, \<sigma>', \<zeta>') \<in> automaton.trans (opnet onp n)"   and "\<forall>j. j \<notin> net_ips \<zeta> \<longrightarrow> \<sigma>' j = \<sigma> j"   and "netgmap sr s' = netmask (net_tree_ips n) (\<sigma>', \<zeta>')" shows "thesis"
vrange_atE: fixes b :: "V" assumes "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> vrat_plus" obtains a :: "V" where "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> vrat_plus" and "vrat_plus\<lparr>a\<rparr> = b"
concurrent_create_delete_independent_technical: fixes i :: "'a"   and "is" :: "'a set"   and xs :: "('a \<times> ('a, 'b) operation) event list"   and j :: "nat"   and e :: "'b"   and ir :: "'a" assumes "i \<in> is"   and "xs prefix of j"   and "(i, Create i e) \<in> set (node_deliver_messages xs)"   and "(ir, Delete is e) \<in> set (node_deliver_messages xs)" shows "hb (i, Create i e) (ir, Delete is e)"
r_inv: fixes x :: "'a" assumes "x \<in> carrier H'" shows "x \<otimes>\<^bsub>H'\<^esub> inv\<^bsub>H'\<^esub> x = \<one>\<^bsub>H'\<^esub>"
nat_pow_consistent: fixes x :: "'a"   and n :: "nat"   and H :: "'a set" shows "x [^] n = x [^]\<^bsub>R\<lparr>carrier := H\<rparr>\<^esub> n"
set_zip_leftD: fixes x :: "'c"   and y :: "'d"   and xs :: "'c list"   and ys :: "'d list" assumes "(x, y) \<in> set (zip xs ys)" shows "x \<in> set xs"