hcju/isabelleps huggingface dataset:
univ_commute: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and f :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "equiv A r"   and "f respects r"   and "x \<in> A" shows "univ f (Equiv_Relations.proj r x) = f x"
the_inv_into_in_Pi: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" assumes "inj_on f A" shows "the_inv_into A f \<in> f ` A \<rightarrow> A"
A_2: shows "typ_of f = Some (\<tau> \<rightarrow> \<tau>')"
isomorphic_integer_zeroth_homology_group: fixes X :: "'a topology" assumes "path_connected_space X"   and "topspace X \<noteq> {}" shows "homology_group 0 X \<cong> integer_group"
elims_3: fixes \<delta> :: "'e \<Rightarrow> 'f \<times> 'g \<Rightarrow> 'e"   and \<gamma> :: "'e \<Rightarrow> 'f \<times> 'g \<Rightarrow> 'h"   and bot :: "'e"   and id1 :: "nat"   and id2 :: "nat"   and x :: "('i, 'j, 'k, 'l) SLin_state_scheme \<times> ('m, 'j, 'k, 'n) SLin_state_scheme" assumes "Idempotence \<delta> \<gamma> bot id1 id2"   and "\<not> P7 x" obtains s1 :: "('i, 'j, 'k, 'l) SLin_state_scheme" and s2 :: "('m, 'j, 'k, 'n) SLin_state_scheme" where "x = (s1, s2)" and "\<not> (\<forall>c. status s1 c = Aborted \<and> \<not> initialized s2 \<longrightarrow> pending s2 c = pending s1 c \<and> status s2 c \<in> {Pending, Aborted})"
subgraph_def: fixes H :: "('e, 'f) pre_digraph"   and G :: "('e, 'f) pre_digraph" shows "subgraph H G \<equiv> verts H \<subseteq> verts G \<and> arcs H \<subseteq> arcs G \<and> wf_digraph G \<and> wf_digraph H \<and> compatible G H"
h_in_hs: fixes as :: "real list"   and bs :: "real list"   and hs :: "(real \<Rightarrow> real) list"   and k :: "nat"   and x\<^sub>0 :: "real"   and x\<^sub>1 :: "real"   and hb :: "real"   and e :: "real"   and p :: "real"   and i :: "nat" assumes "akra_bazzi_real_recursion as bs hs k x\<^sub>0 x\<^sub>1 hb e p"   and "i < k" shows "hs ! i \<in> set hs"
replace_append: fixes x :: "'a"   and ys :: "'a list"   and as :: "'a list"   and bs :: "'a list" shows "replace x ys (as @ bs) = (if x \<in> set as then replace x ys as @ bs else as @ replace x ys bs)"
opp_mult_def: fixes x :: "'a"   and y :: "'a" shows "x \<odot> y \<equiv> y \<cdot> x"
convex_set_plus: fixes S :: "'a set"   and T :: "'a set" assumes "convex S"   and "convex T" shows "convex (S + T)"
primes_M_def: shows "primes_M = prime_sum_upto (\<lambda>p. ln (real p) / real p)"
select_convs_2: fixes Calculus :: "'a processCalculus"   and HasBarb :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and more :: "'c" shows "HasBarb \<lparr>Calculus = Calculus, HasBarb = HasBarb, \<dots> = more\<rparr> = HasBarb"
RedFAccNull: fixes E :: "char list \<Rightarrow> ty option"   and F :: "char list"   and Cs :: "char list list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list" shows "P,E \<turnstile> \<langle>null\<bullet>F{Cs},s\<rangle> \<rightarrow> \<langle>Throw (addr_of_sys_xcpt NullPointer, [NullPointer]),s\<rangle>"
derivable_premisses: fixes C :: "'a eclause"   and P :: "'a eclause set"   and S :: "'a eclause set"   and \<sigma> :: "('a \<times> 'a trm) list"   and k :: "inferences"   and C' :: "'a literal set" assumes "derivable C P S \<sigma> k C'" shows "P \<subseteq> S"
Alloc_def: fixes G :: "prog"   and otag :: "obj_tag"   and P :: "'b assn" shows "Alloc G otag P = (\<lambda>Y s Z. \<forall>s' a. G\<turnstile>s \<midarrow>halloc otag\<succ>a\<rightarrow> s' \<longrightarrow> P (In1 (Addr a)) s' Z)"
m_translate_functions: fixes c :: "'a" assumes "c \<in> carrier R" shows "m_translate c \<in> carrier (function_ring (carrier R) R)"
nat_mono_iff: fixes z :: "int"   and w :: "int" assumes "0 < z" shows "(nat w < nat z) = (w < z)"
zcf_one: shows "zcf \<one>\<^bsub>P\<^esub> = \<one>"
upper_dimension_axiom: fixes x :: "complex_homo"   and y :: "complex_homo"   and z :: "complex_homo"   and u :: "complex_homo"   and v :: "complex_homo" assumes "x \<in> unit_disc"   and "y \<in> unit_disc"   and "z \<in> unit_disc"   and "u \<in> unit_disc"   and "v \<in> unit_disc"   and "poincare_distance x u = poincare_distance x v"   and "poincare_distance y u = poincare_distance y v"   and "poincare_distance z u = poincare_distance z v"   and "u \<noteq> v" shows "poincare_between x y z \<or> poincare_between y z x \<or> poincare_between z x y"
loc_uncons: fixes a :: "'a el_step"   and b :: "'a el_step list" shows "loc (a # b) = (case a of LOC e \<Rightarrow> [a] | ENV e \<Rightarrow> []) @ loc b"
Inv_VcompNml: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Can t"   and "Can u"   and "Nml t"   and "Nml u"   and "Dom t = Cod u" shows "Inv (t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> u) = Inv u \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> Inv t"
less_eq_st_rep_def: fixes ps1 :: "(char list \<times> 'a) list"   and ps2 :: "(char list \<times> 'a) list" shows "local.less_eq_st_rep ps1 ps2 = (\<forall>x\<in>set (map fst ps1) \<union> set (map fst ps2). fun_rep ps1 x \<le> fun_rep ps2 x)"
no_volatile_Write\<^sub>s\<^sub>b: shows "outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb = {}"
affine_dependent_explicit_finite: fixes S :: "'a set" assumes "finite S" shows "affine_dependent S = (\<exists>u. sum u S = 0 \<and> (\<exists>v\<in>S. u v \<noteq> 0) \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = (0::'a))"
nextmem_exists_and_unique: fixes c :: "('exp, 'id) MWLsCom"   and m :: "'id \<Rightarrow> 'val" shows "\<exists>m' p \<alpha>. SRabbr c m \<alpha> p m' \<and> (\<forall>m''. (\<exists>p \<alpha>. SRabbr c m \<alpha> p m'') \<longrightarrow> m'' = m')"
is_SupI: fixes A :: "'b set"   and sup :: "'b" assumes "\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> sup"   and "\<And>z. \<forall>x\<in>A. x \<sqsubseteq> z \<Longrightarrow> sup \<sqsubseteq> z" shows "is_Sup A sup"
compact_interval: fixes a :: "'a"   and b :: "'a" shows "compact {a..b}"
that: fixes x :: "'d" assumes "h \<turnstile> f \<rightarrow>\<^sub>r x"   and "h \<turnstile> g x \<rightarrow>\<^sub>h h'" shows "thesis"
numeral_One: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a" assumes "class.numeral plus" shows "numeral.numeral one plus num.One = one"
unitary11_gen_regular: fixes M :: "complex \<times> complex \<times> complex \<times> complex" assumes "unitary11_gen M" shows "mat_det M \<noteq> 0"
Union_paths_carrier: fixes t :: "'b ttree" shows "\<Union> (set ` paths t) = carrier t"
set_insert: fixes x :: "'a"   and A :: "'a set" assumes "x \<in> A" obtains B :: "'a set" where "A = {x} \<union> B" and "x \<notin> B"
presWlsAll_fromIMorAll: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<times> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<times> 'd"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "ipresWlsAll h hA (fromMOD MOD)" shows "presWlsAll (fromIMor h) (fromIMorAbs hA) MOD"
norm_def: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'b DBMEntry"   and k :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "DBM_Normalization.norm M k n \<equiv> \<lambda>i j. let ub = if 0 < i then k i else (0::'b); lb = if 0 < j then - k j else (0::'b) in if i \<le> n \<and> j \<le> n then norm_lower (norm_upper (M i j) ub) lb else M i j"
induct: fixes x :: "'g set"   and P :: "'g set \<Rightarrow> bool" assumes "finite x"   and "P {}"   and "\<And>A a. \<lbrakk>finite A; P A\<rbrakk> \<Longrightarrow> P (insert a A)" shows "P x"
simps_3: fixes x :: "(int \<times> int \<times> protocol) set set"   and y :: "(int \<times> int \<times> protocol) set set"   and p :: "protocol \<times> int" shows "Cp (AllowPortFromTo x y p) = allow_from_to_port_prot (fst p) (snd p) x y"
or_int_rec: fixes k :: "int"   and l :: "int" shows "k OR l = of_bool (odd k \<or> odd l) + 2 * (k div 2 OR l div 2)"
hyps': fixes x :: "real" shows "real (Suc n) * bernpoly n (x + 1) - real (Suc n) * bernpoly n x = real n * real (Suc n) * x ^ (n - Suc 0)"
Rep_af_inject: fixes x :: "'d af"   and y :: "'d af" shows "(??.Abstract_AF.af.Rep_af x = ??.Abstract_AF.af.Rep_af y) = (x = y)"
SUP_LE_P_def: fixes X :: "'a \<Rightarrow> 'b \<Rightarrow> 'e"   and u :: "'c"   and i :: "'b" shows "SUP_LE_P X u i = (SUP v\<in>{v. pair v i \<sqsubseteq> u}. X v i)"
H: fixes c :: "real"   and f :: "'a" shows "N (c *\<^sub>R f) \<le> ennreal \<bar>c\<bar> * N f"
\<phi>_uniqueness_2: shows "local.iso t\<^sub>0u\<^sub>1.\<phi>"
Always_weaken: fixes F :: "'d program"   and A :: "'d set"   and B :: "'d set" assumes "F \<in> Always A"   and "A \<subseteq> B" shows "F \<in> Always B"
partn_lst_def: fixes r :: "('b \<times> 'b) set"   and B :: "'b set"   and \<alpha> :: "V list"   and n :: "nat" shows "Partitions.partn_lst r B \<alpha> n \<equiv> \<forall>f\<in>[B]\<^bsup>n\<^esup> \<rightarrow> {..<length \<alpha>}. \<exists>i<length \<alpha>. \<exists>H\<subseteq>B. ordertype H r = \<alpha> ! i \<and> f ` [H]\<^bsup>n\<^esup> \<subseteq> {i}"
pmaxsimpchain_Cons_reduce: fixes X :: "'b set set"   and x :: "'b set"   and xs :: "'b set list" assumes "SimplicialComplex X"   and "SimplicialComplex.pmaxsimpchain X (x # xs)" shows "SimplicialComplex.pmaxsimpchain X xs"
ev_1: shows "\<forall>\<^sub>F n in F. l \<le> f n / g n"
ideal_generated_singleton_subset: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and d :: "'b"   and S :: "'b set" assumes "class.comm_ring_1 times one plus zero minus uminus"   and "d \<in> ring.ideal_generated plus zero uminus times S"   and "finite S" shows "ring.ideal_generated plus zero uminus times {d} \<subseteq> ring.ideal_generated plus zero uminus times S"
p_Suc: fixes x :: "'s"   and y :: "'s"   and n :: "nat" shows "ennreal (p x y (Suc n)) = \<integral>\<^sup>+ x. ennreal (p x y n) \<partial>measure_pmf (K x)"
assms_2: shows "group_homomorphism \<theta> B multB oneB C multC oneC"
\<epsilon>_in_hom: shows "\<guillemotleft>\<xi> \<cdot> (f \<star> local.inv \<eta> \<star> g) \<cdot> (local.inv \<xi> \<star> f \<star> g) : f \<star> g \<Rightarrow> trg f\<guillemotright>"
simps_1: fixes A :: "'a iarray iarray"   and i :: "nat"   and j :: "nat"   and res :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" shows "Hermite_reduce_above_iarrays A 0 i j res = A"
behaviour_refinement: fixes R :: "('a \<times> 'b) set"   and Ta :: "('a, 'c) TS_scheme"   and Tc :: "('b, 'd) TS_scheme"   and bc :: "'b list" assumes "PO_refines R Ta Tc"   and "bc \<in> beh Tc" shows "\<exists>ba\<in>beh Ta. (ba, bc) \<in> seq_lift R"
intro: fixes f :: "nat \<Rightarrow> 'a" assumes "f 0 = (0::'a)"   and "f (Suc 0) = (1::'a)"   and "\<And>a b. \<lbrakk>1 < a; 1 < b\<rbrakk> \<Longrightarrow> f (a * b) = f a * f b" shows "completely_multiplicative_function f"
Id_on_domain: fixes A :: "'a set"   and f :: "('a \<times> 'b) set" shows "Domain (Id_on A O f) = A \<inter> Domain f"
simps_8: fixes f :: "'c \<Rightarrow> 'b" shows "map_Tree f Tip = Tip"
cobounded1: fixes a :: "'a"   and b :: "'a" shows "a \<le> sup a b"
IH: shows "s = istate \<or> (\<exists>tr. valid tr \<and> srcOf (hd tr) = istate \<and> tgtOf (last tr) = s)"
difference_def: fixes xs :: "'b odlist"   and ys :: "'b odlist" shows "ODList.difference xs ys = ODList (List_local.difference (toList xs) (toList ys))"
Exit_no_sum_SDG_edge_target: fixes n :: "'node SDG_node"   and Vopt :: "'var option"   and popt :: "('pname \<times> bool) option"   and b :: "bool" assumes "sum_SDG_edge n Vopt popt b (CFG_node (_Exit_))" shows "False"
assert_spmf_simps_1: shows "assert_spmf True = return_spmf ()"
poincare_between_u0v: fixes u :: "complex_homo"   and v :: "complex_homo" assumes "u \<in> unit_disc"   and "v \<in> unit_disc"   and "u \<noteq> 0\<^sub>h"   and "v \<noteq> 0\<^sub>h" shows "poincare_between u 0\<^sub>h v = (\<exists>k<0. to_complex u = cor k * to_complex v)"
frechet_derivative_zero_fun: fixes a :: "'a" shows "frechet_derivative 0 (at a) = 0"
simps_10: fixes t :: "'c local.term"   and u :: "'c local.term"   and v :: "'c local.term" shows "Inv \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[t, u, v\<^bold>] = \<^bold>\<a>\<^bold>[Inv t, Inv u, Inv v\<^bold>]"
abs_summable_on_altdef: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set" shows "(f abs_summable_on A) = set_integrable (count_space UNIV) A f"
U_LIMCONS_in_Lim: shows "U\<^bsub>LIM-CONS\<^esub> \<in> LIM"
welltypedE_3: fixes b :: "bool"   and T :: "Unary_PCF.type" assumes "b ::: T" obtains "T = \<B>"
Rats_add: fixes a :: "'b"   and b :: "'b" assumes "a \<in> \<rat>"   and "b \<in> \<rat>" shows "a + b \<in> \<rat>"
minus_plus_minus_cancel: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and u :: "'b"   and t :: "'b"   and s :: "'b" assumes "class.comm_powerprod plus minus zero"   and "plus.adds plus u t"   and "plus.adds plus s u" shows "plus (minus t u) (minus u s) = minus t s"
ConstFunctor'_def: fixes A :: "('a, 'c, 'e) Category_scheme"   and B :: "('b, 'd, 'f) Category_scheme"   and b :: "'b" shows "ConstFunctor' A B b \<equiv> \<lparr>CatDom = A, CatCod = B, MapM = \<lambda>f. id\<^bsub>B\<^esub> b\<rparr>"
conc_trans_additional_4: fixes A :: "'a nres"   and B :: "'a nres"   and C :: "'a nres" assumes "A \<le> \<Down> Id B"   and "B \<le> \<Down> Id C" shows "A \<le> C"
filterlim_tendsto_neg_mult_at_bot: fixes f :: "'b \<Rightarrow> real"   and c :: "real"   and F :: "'b filter"   and g :: "'b \<Rightarrow> real" assumes "(f \<longlongrightarrow> c) F"   and "c < 0"   and "filterlim g at_top F" shows "LIM x F. f x * g x :> at_bot"
progressing_single: fixes e :: "'b \<times> 'b" shows "progressing [e]"
distincts_inj_on_set: fixes xss :: "'b list list" assumes "distincts xss" shows "inj_on set (set xss)"
symm_closure_of_indRelR_1: shows "indRel = symcl indRelR"
Convm_def: shows "Convm = Conv (copy_first_matrix nr (output_size' m)) m"
refl_on_domain: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and a :: "'a"   and b :: "'a" assumes "refl_on A r"   and "(a, b) \<in> r" shows "a \<in> A \<and> b \<in> A"
homE_2: fixes A :: "'o"   and B :: "'o"   and f :: "'a" assumes "A \<in> Ob"   and "B \<in> Ob"   and "f \<in> Hom A B" shows "Cod f = B"
sorted_wrt01: fixes xs :: "'g list"   and P :: "'g \<Rightarrow> 'g \<Rightarrow> bool" assumes "length xs \<le> 1" shows "sorted_wrt P xs"
add_left_mono: fixes a :: "'b multiset"   and b :: "'b multiset"   and c :: "'b multiset" assumes "a \<subseteq># b" shows "c + a \<subseteq># c + b"
Dom_SPN: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and \<mu> :: "'b" assumes "bicategory_of_spans V H \<a> \<i> src trg"   and "partial_magma.arr V \<mu>" shows "Dom (bicategory_of_spans.SPN V H \<a> \<i> src trg \<mu>) = \<lparr>Leg0 = Maps.MkArr (src (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (partial_magma.dom V \<mu>))) (trg (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (partial_magma.dom V \<mu>))) (category.iso_class V (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (partial_magma.dom V \<mu>))), Leg1 = Maps.MkArr (src (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg (partial_magma.dom V \<mu>))) (trg (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg (partial_magma.dom V \<mu>))) (category.iso_class V (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg (partial_magma.dom V \<mu>)))\<rparr>"
has_contour_integral_def: fixes f :: "complex \<Rightarrow> complex"   and i :: "complex"   and g :: "real \<Rightarrow> complex" shows "(f has_contour_integral i) g \<equiv> ((\<lambda>x. f (g x) * vector_derivative g (at x within {0..1})) has_integral i) {0..1}"
face_distance_def: fixes F :: "'a set"   and D :: "'a set" shows "face_distance F D \<equiv> chamber_distance (closest_supchamber F D) D"
assms_2: fixes a :: "'a"   and b :: "'b" assumes "R a b" shows "rel_option R (f a) (g b)"
length_cons_code: fixes g :: "('e, 'f) generator"   and s :: "'f" shows "length_cons g s = (case generator g s of Done \<Rightarrow> 0 | Skip s' \<Rightarrow> length_cons g s' | Yield a s' \<Rightarrow> 1 + length_cons g s')"
l_2: shows "(\<lambda>n. to_Gromov_completion (v n)) \<longlonglongrightarrow> l"
F2set1_bd': fixes x :: "('d, 'e, 'f) F2"   and r1 :: "(bd_type_F1 \<times> bd_type_F1) set" shows "|F2set1 x| \<le>o dir_image (r1 +c bd_F2) Abs_bd_type_F"
wf_trm_subterm: fixes t :: "('fun, 'a) Term.term"   and s :: "('fun, 'a) Term.term" assumes "wf\<^sub>t\<^sub>r\<^sub>m t"   and "s \<sqsubset> t" shows "wf\<^sub>t\<^sub>r\<^sub>m s"
Cons_1: fixes s :: "'state" assumes "exit s (hd vl)"   and "validFrom s tr"   and "V tr = vl"   and "never T tr" shows "False"
univ_cmp_def: fixes x :: "'b"   and y :: "'b" shows "univ_cmp x y \<equiv> if x = y then EQUAL else if (x, y) \<in> univ_order_rel then LESS else GREATER"
subgroup_subgroup_generated_iff: fixes G :: "('f, 'g) monoid_scheme"   and H :: "'f set"   and B :: "'f set" assumes "Group.group G" shows "subgroup H (subgroup_generated G B) = (subgroup H G \<and> H \<subseteq> carrier (subgroup_generated G B))"
WW_seg_subset_WW: fixes n :: "nat" shows "WW_seg n \<subseteq> WW"
restrict0_in_fun_space: fixes charts :: "('c, 'd) chart set"   and k :: "enat"   and f :: "'c \<Rightarrow> real" assumes "c_manifold charts k"   and "diff_fun k charts f" shows "restrict0 (manifold.carrier charts) f \<in> c_manifold.diff_fun_space charts k"
size_heap_of_A: fixes xs :: "'b list" shows "size (heap_of_A xs) = length xs"
row_space_def: fixes n :: "nat"   and B :: "'a mat" shows "vec_space.row_space n B = LinearCombinations.module.span class_ring (module_vec TYPE('a) n) (set (rows B))"
mult_less_cancel_left_pos: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and c :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times"   and "less zero c" shows "less (times c a) (times c b) = less a b"
id4: shows "(xi, xsj, smult (inverse (xsj - xi)) ([:- xi, 1::'a:] * p_sisj + [:xsj, - (1::'a):] * p_ij)) = (x l, x (l + Suc j), local.neville_aitken_main l (l + Suc j))"
ccApprox_both: fixes t :: "Vars.var ttree"   and t' :: "Vars.var ttree" shows "ccApprox (t \<otimes>\<otimes> t') = ccApprox t \<squnion> ccApprox t' \<squnion> carrier t G\<times> carrier t'"
distinct_1: fixes subject :: "subject"   and name :: "name"   and pi :: "pi"   and freeRes' :: "freeRes"   and pi' :: "pi" shows "subject\<guillemotleft>name\<guillemotright> \<prec> pi \<noteq> freeRes' \<prec> pi'"
content_cball: fixes r :: "real"   and c :: "'b" assumes "0 \<le> r" shows "content (cball c r) = unit_ball_vol (real DIM('b)) * r ^ DIM('b)"
given_clause_basis_axioms: shows "given_clause_basis Bot_F Inf_F Bot_G Q entails_q Inf_G_q Red_I_q Red_F_q \<G>_F_q \<G>_I_q Inf_FL (\<doteq>) (\<prec>\<cdot>) (\<sqsubset>L) active"
pcr_cblinfun_def: fixes T :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and Ta :: "'b \<Rightarrow> 'd \<Rightarrow> bool" shows "pcr_cblinfun T Ta \<equiv> rel_fun T Ta OO cr_cblinfun"
hides_def: fixes G :: "prog"   and new :: "qtname \<times> mdecl"   and old :: "qtname \<times> mdecl" shows "G\<turnstile> new hides old = (is_static new \<and> msig new = msig old \<and> G\<turnstile>declclass new\<prec>\<^sub>C declclass old \<and> G\<turnstile>Method new declared_in declclass new \<and> G\<turnstile>Method old declared_in declclass old \<and> G \<turnstile>Method old inheritable_in pid (declclass new) )"
binomial_ngt_diff: fixes n :: "nat" shows "n choose Suc n = Suc n choose Suc n - (n choose n)"
assms_5: shows "F\<^sub>i\<^sub>n\<^sub>f = (\<forall>rel\<in>R. \<forall>(a, b, c)\<in>Atoms rel. \<exists>x. \<forall>y<x. rel (a * y\<^sup>2 + b * y + c) 0)"
assms: fixes g :: "'g" assumes "g \<in> G" shows "T (T g) = T g"
preserves_modes_mem_def2: fixes \<R> :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) \<times> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) set" shows "preserves_modes_mem \<R> = (\<forall>c\<^sub>A mds\<^sub>A mem\<^sub>A c\<^sub>C mds\<^sub>C mem\<^sub>C. (\<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>A\<rangle>\<^sub>A, \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C\<rangle>\<^sub>C) \<in> \<R> \<longrightarrow> mem\<^sub>A = mem\<^sub>A_of mem\<^sub>C \<and> mds\<^sub>A = mds\<^sub>A_of mds\<^sub>C)"
inner_node_Entry_edge: fixes n :: "'b" assumes "inner_node n" obtains a :: "'a" where "valid_edge a" and "inner_node (targetnode a)" and "sourcenode a = Entry"
Abs_ordinal_inject: fixes x :: "ord0 set"   and y :: "ord0 set" assumes "x \<in> UNIV // ord0rel"   and "y \<in> UNIV // ord0rel" shows "(Abs_ordinal x = Abs_ordinal y) = (x = y)"
theorem24xxiii: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> y \<sqinter> - (x \<sqinter> z) = x \<sqinter> (y \<sqinter> - z) \<sqinter> - (x \<sqinter> z)"
Tau: shows "\<T> = generate_valid_stateful_policy_IFSACS G M edgesList"
card_Diff1_less: fixes A :: "'b set"   and x :: "'b" assumes "finite A"   and "x \<in> A" shows "card (A - {x}) < card A"
problem_plan_bound_s_works: fixes PROB :: "(('c, bool) fmap \<times> ('c, bool) fmap) set"   and as :: "(('c, bool) fmap \<times> ('c, bool) fmap) list"   and s :: "('c, bool) fmap" assumes "finite PROB"   and "as \<in> valid_plans PROB"   and "s \<in> valid_states PROB" shows "\<exists>as'. exec_plan s as = exec_plan s as' \<and> subseq as' as \<and> length as' \<le> problem_plan_bound_s PROB s"
bdd_above_def: fixes A :: "'a multiset set" shows "subset_mset.bdd_above A = (\<exists>M. \<forall>x\<in>A. x \<subseteq># M)"
continuous_on_transfer_right_total2: fixes B :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and BP :: "'b \<Rightarrow> bool"   and A :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "Domainp B = BP"   and "bi_unique A"   and "bi_total A"   and "rel_fun (rel_set A) (=) open open"   and "bi_unique B"   and "right_total B"   and "rel_fun (rel_set B) (=) (openin (top_of_set (Collect BP))) open" shows "rel_fun (rel_set A) (rel_fun (rel_fun A B) (=)) continuous_on continuous_on"
union_diff_assoc: fixes C :: "'c multiset"   and B :: "'c multiset"   and A :: "'c multiset" assumes "C - B = {#}" shows "A + B - C = A + (B - C)"
indets_monomial: fixes c :: "'a"   and t :: "'b \<Rightarrow>\<^sub>0 nat" assumes "c \<noteq> (0::'a)" shows "indets (monomial c t) = keys t"
ipresCons_defs_1: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'e"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'f"   and MOD :: "('index, 'bindex, 'varSort, 'g, 'opSym, 'var, 'e, 'f, 'h) model_scheme" shows "ipresCons h hA MOD \<equiv> ipresVar h MOD \<and> ipresAbs h hA MOD \<and> ipresOp h hA MOD"
simps_2: fixes uu :: "int \<Rightarrow> int \<Rightarrow> bool"   and uw :: "value option" shows "MaybeBoolInt uu None uw = invalid"
Nfunc_LeastTr2_6: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a set"   and g :: "nat \<Rightarrow> 'a set" assumes "\<forall>l\<le>n. G \<guillemotright> f l "   and "\<forall>l\<le>n. G \<guillemotright> g l "   and "\<exists>i\<le>n. f i \<noteq> g i" shows "(LEAST k. f k \<noteq> g k) \<le> n"
lim_mono: fixes N :: "nat"   and X :: "nat \<Rightarrow> 'c"   and Y :: "nat \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "\<And>n. N \<le> n \<Longrightarrow> X n \<le> Y n"   and "X \<longlonglongrightarrow> x"   and "Y \<longlonglongrightarrow> y" shows "x \<le> y"