hcju/isabelleps huggingface dataset:
LIMSEQ_NSLIMSEQ: fixes X :: "nat \<Rightarrow> 'a"   and L :: "'a" assumes "X \<longlonglongrightarrow> L" shows "X \<longlonglongrightarrow>\<^sub>N\<^sub>S L"
axioms_2: fixes \<alpha> :: "V"   and \<CC> :: "V" assumes "digraph \<alpha> \<CC>" shows "vfsequence \<CC>"
ord_Zp_not_equal_imp_notequal_3: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> nonzero Zp"   and "y \<in> nonzero Zp"   and "ord_Zp x \<noteq> ord_Zp y" shows "x \<oplus> y \<noteq> \<zero>"
det_fold_map_def: fixes R :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and c :: "'g \<Rightarrow> bool"   and f :: "'f \<times> 'h \<Rightarrow> 'g \<Rightarrow> 'g"   and \<sigma> :: "'g"   and result :: "('f \<Rightarrow> 'h option) \<Rightarrow> 'g" shows "det_fold_map R c f \<sigma> result \<equiv> \<forall>l. distinct (map fst l) \<and> sorted_wrt (key_rel R) l \<longrightarrow> foldli l c f \<sigma> = result (map_of l)"
simps_6: fixes I :: "('e, 'f, 'd) interp"   and \<alpha> :: "('e, 'f, 'd) hp"   and \<phi> :: "('e, 'f, 'd) formula" shows "fml_sem I (\<langle> \<alpha> \<rangle> \<phi>) = {uu_. \<exists>\<nu> \<omega>. uu_ = \<nu> \<and> (\<nu>, \<omega>) \<in> prog_sem I \<alpha> \<and> \<omega> \<in> fml_sem I \<phi>}"
intro: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and sep_size :: "nat"   and P_new :: "'c list" assumes "\<And>paths' P_new'. ProofStepInduct G v0 v1 paths' P_new' sep_size \<Longrightarrow> Digraph.distance (Digraph.remove_vertex G v0) (last P_new) v1 \<le> Digraph.distance (Digraph.remove_vertex G v0) (last P_new') v1" shows "ProofStepInductOptimalPaths_axioms G v0 v1 P_new sep_size"
SCAs0_to_SMCAs: shows "map S CAs0 \<cdot>\<cdot>cl \<eta>s0 = map (S_M S M) CAs"
finite_imp_finite_component_Keys: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "finite F" shows "finite (component_of_term ` Keys F)"
IH: fixes xs :: "'a llist" assumes "zs \<sqsubseteq> xs" shows "lSup {ys. ys \<sqsubseteq> xs \<and> zs \<sqsubseteq> ys \<and> lfinite ys} = xs"
assms_4: shows "y \<noteq> 0\<^sub>h"
alphaBinp_trans: fixes qbinp :: "'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qAbs option"   and qbinp' :: "'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qAbs option"   and qbinp'' :: "'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qAbs option" assumes "qGoodBinp qbinp"   and "(qbinp, qbinp') \<in> alphaBinp"   and "(qbinp', qbinp'') \<in> alphaBinp" shows "(qbinp, qbinp'') \<in> alphaBinp"
proj_stoch_set_discriminating: fixes p :: "real"   and M :: "bool stream measure"   and F :: "nat \<Rightarrow> bool stream measure"   and X :: "nat \<Rightarrow> bool stream \<Rightarrow> 'b"   and n :: "nat"   and N :: "'b stream measure" assumes "infinite_cts_filtration p M F"   and "borel_adapt_stoch_proc F X" shows "set_discriminating n (proj_stoch_proc X n) N"
size_pass\<^sub>1: fixes hp :: "'b tree" shows "size (pass\<^sub>1 hp) = size hp"
invimage_vrrestriction: fixes r :: "V"   and A :: "V"   and B :: "V" shows "(r \<restriction>\<^sup>r\<^sub>\<circ> A) -`\<^sub>\<circ> B = r -`\<^sub>\<circ> (A \<inter>\<^sub>\<circ> B)"
not_less_iff_gr_or_eq: fixes y :: "'a"   and x :: "'a" shows "(\<not> y < x) = (x < y \<or> x = y)"
less_eq_complex_lex_def: fixes x :: "complex"   and y :: "complex" shows "(x \<le>\<^sub>\<complex> y) = (Re x < Re y \<or> Re x = Re y \<and> Im x \<le> Im y)"
map_id: fixes t :: "'a dnelist" shows "dnelist.map id t = t"
assms_13: shows "r * c = p * (r + c + u) + p + u"
lappend_assoc: fixes xs :: "'b llist"   and ys :: "'b llist"   and zs :: "'b llist" shows "(xs :++ ys) :++ zs = xs :++ ys :++ zs"
IH: shows "lookup (monomial c t * except p {w}) s = (c * lookup (except p {w}) (s - t) when t adds s)"
op_dg_components_4: fixes \<CC> :: "V" shows "op_dg \<CC>\<lparr>Cod\<rparr> = \<CC>\<lparr>Dom\<rparr>"
unit_char_1: fixes a :: "'c" assumes "C.obj a" shows "\<guillemotleft>unit a : map\<^sub>0 a \<Rightarrow>\<^sub>D F a\<guillemotright>"
exhaust: fixes y :: "bit" obtains "y = \<zero>"   | "y = \<one>"
coherent_Runit_Ide: fixes a :: "'c local.term" assumes "Ide a" shows "coherent \<^bold>\<r>\<^bold>[a\<^bold>]"
some_eq_ex: fixes P :: "'c \<Rightarrow> bool" shows "P (SOME x. P x) = (\<exists>x. P x)"
foldl_set: fixes l :: "'d set list" shows "foldl (\<union>) {} l = \<Union> {x. x \<in> set l}"
ppvs_decomp: shows "ppvs = fppvs @ bppvs"
rbt_foldi_fold_conj: fixes P :: "'b \<Rightarrow> bool"   and t :: "('b, 'c) RBT_Impl.rbt"   and val :: "bool" shows "foldi (\<lambda>s. s = True) (\<lambda>k v s. s \<and> P k) t val = RBT_Impl.fold (\<lambda>k v s. s \<and> P k) t val"
NEQ_DEP_IMP_IN_DOM_ii: fixes a :: "('d, 'e) fmap \<times> ('d, 'f) fmap"   and PROB :: "(('d, 'e) fmap \<times> ('d, 'f) fmap) set"   and v :: "'d" assumes "a \<in> PROB"   and "v \<in> fmdom' (snd a)" shows "v \<in> prob_dom PROB"
innermost_intvl2: fixes f :: "atom fm"   and xs :: "real list"   and l :: "real"   and x :: "real"   and y :: "real" assumes "nqfree f"   and "nolb f xs l x"   and "l < x"   and "x \<notin> EQ2 f xs"   and "R.I f (x # xs)"   and "l < y"   and "y \<le> x" shows "R.I f (y # xs)"
butlast_rest_bl2bin: fixes bl :: "bool list" shows "bl_to_bin (butlast bl) = bl_to_bin bl div 2"
xor_bool_def: fixes x :: "bool"   and y :: "bool" shows "x \<oplus> y = (x \<noteq> y)"
cont_int: shows "continuous_on {a..b} (\<lambda>x. integral {c..g x} f)"
simps_2: fixes n :: "C"   and ns :: "C list" shows "C2O (C (n # ns)) = addO (C2O (C ns)) (exp\<omega> (C2O n))"
DBM_canonical_subset_le: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry"   and M' :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry"   and i :: "nat"   and j :: "nat" assumes "canonical M n"   and "[M]\<^bsub>v,n\<^esub> \<subseteq> [M']\<^bsub>v,n\<^esub>"   and "[M]\<^bsub>v,n\<^esub> \<noteq> {}"   and "i \<le> n"   and "j \<le> n"   and "i \<noteq> j" shows "M i j \<le> M' i j"
prv_eq_isTrue: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "prv \<phi> = isTrue \<phi>"
stationary_distribution_def: fixes N :: "'s spmf" shows "K2.stationary_distribution N = (N = N \<bind> K')"
lead_coeff_normalize_field: fixes p :: "'b poly" assumes "p \<noteq> 0" shows "lead_coeff (normalize p) = (1::'b)"
fresh_def: fixes a :: "'e"   and x :: "'f" shows "a \<sharp> x = (a \<notin> supp x)"
refl: fixes cmp :: "'b \<Rightarrow> 'b \<Rightarrow> comp"   and a :: "'b" assumes "comparator cmp" shows "cmp a a = Equiv"
C: shows "a \<otimes>\<^bsub>padic_int p\<^esub> b = \<zero>\<^bsub>padic_int p\<^esub>"
assms_2: shows "convert_atom_list var L2 (xs' @ x # xs) = Some L2'"
eventually_at_left: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and "open" :: "'b set \<Rightarrow> bool"   and y :: "'b"   and x :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "class.linorder_topology less_eq less open"   and "less y x" shows "eventually P (topological_space.at_within open x (ord.lessThan less x)) = (\<exists>b. less b x \<and> (\<forall>y. less b y \<longrightarrow> less y x \<longrightarrow> P y))"
\<Delta>'''_sel_1: fixes x :: "'v"   and y :: "'v" shows "edge \<Delta>''' x y = (edge \<Delta> x y \<and> 0 < capacity \<Delta> (x, y) \<and> y \<noteq> source \<Delta> \<and> x \<noteq> sink \<Delta>)"
Uhyp: fixes s :: "'a" assumes "s \<in> S" shows "is_interval (U s)"
lNAct_ex: fixes c :: "'id"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat" shows "\<langle>c \<Leftarrow> t\<rangle>\<^bsub>n\<^esub> = n \<or> \<langle>c \<Leftarrow> t\<rangle>\<^bsub>n\<^esub> < n \<and> \<not> (\<exists>k\<ge>\<langle>c \<Leftarrow> t\<rangle>\<^bsub>n\<^esub>. k < n \<and> \<parallel>c\<parallel>\<^bsub>t k\<^esub>)"
simps_1: shows "BinomialHeapStruc.rank_invar [] = True"
drop_append: fixes n :: "nat"   and xs :: "'a list"   and ys :: "'a list" shows "drop n (xs @ ys) = drop n xs @ drop (n - |xs|) ys"
pairwise_disjnt_countable_Union: fixes \<N> :: "'b set set" assumes "countable (\<Union> \<N>)"   and "disjoint \<N>" shows "countable \<N>"
Mg': shows "g \<in> state_measure (V \<union> V') \<Gamma> \<rightarrow>\<^sub>M subprob_algebra R"
reindex_cong: fixes l :: "'e \<Rightarrow> 'f"   and B :: "'e set"   and A :: "'f set"   and g :: "'f \<Rightarrow> 'a"   and h :: "'e \<Rightarrow> 'a" assumes "inj_on l B"   and "A = l ` B"   and "\<And>x. x \<in> B \<Longrightarrow> g (l x) = h x" shows "F.F g A = F.F h B"
normalize_gcd_factorial: 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize" shows "normalize (factorial_semiring.gcd_factorial zero times one normalize a b) = factorial_semiring.gcd_factorial zero times one normalize a b"
induct: fixes x1 :: "('fun, 'var) Term.term set"   and x2 :: "('fun, 'var) Term.term"   and P :: "('fun, 'var) Term.term set \<Rightarrow> ('fun, 'var) Term.term \<Rightarrow> bool" assumes "x1 \<turnstile> x2"   and "\<And>t M. t \<in> M \<Longrightarrow> P M t"   and "\<And>T f M. \<lbrakk>length T = arity f; public f; \<And>t. t \<in> set T \<Longrightarrow> M \<turnstile> t; \<And>t. t \<in> set T \<Longrightarrow> P M t\<rbrakk> \<Longrightarrow> P M (Fun f T)"   and "\<And>M t K T t\<^sub>i. \<lbrakk>M \<turnstile> t; P M t; Ana t = (K, T); \<And>k. k \<in> set K \<Longrightarrow> M \<turnstile> k; \<And>k. k \<in> set K \<Longrightarrow> P M k; t\<^sub>i \<in> set T\<rbrakk> \<Longrightarrow> P M t\<^sub>i" shows "P x1 x2"
3: shows "coreflexive (e * top * e\<^sup>T)"
y: fixes u :: "'a" assumes "u \<in> S" shows "dist z y \<le> dist z u"
Standard_mult: fixes x :: "'a star"   and y :: "'a star" assumes "x \<in> Standard"   and "y \<in> Standard" shows "x * y \<in> Standard"
cases: fixes a1 :: "'a measure"   and a2 :: "'a measure" assumes "a1 \<le> a2" obtains M :: "'a measure" and N :: "'a measure" where "a1 = Ma__" and "a2 = Na__" and "space Ma__ \<subset> space Na__"   | M :: "'a measure" and N :: "'a measure" where "a1 = Mb__" and "a2 = Nb__" and "space Mb__ = space Nb__" and "sets Mb__ \<subset> sets Nb__"   | M :: "'a measure" and N :: "'a measure" where "a1 = M" and "a2 = N" and "space M = space N" and "sets M = sets N" and "emeasure M \<le> emeasure N"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) (Dom t) (Cod t) (Map t)"
cases: fixes x :: "'a rexp" obtains "x = Zero"   | "x = One"   | a :: "'a" where "x = Atom a"   | r1 :: "'a rexp" and r2 :: "'a rexp" where "x = Plus r1__ r2__"   | r1 :: "'a rexp" and r2 :: "'a rexp" where "x = Times r1a__ r2"   | r1 :: "'a rexp" where "x = Star r1"
splus_def: fixes t :: "'a"   and v :: "'a \<times> 'b" shows "pprod.splus t v = (t + pprod.pp_of_term v, pprod.component_of_term v)"
distinct_5: fixes x21 :: "(Term_Class.term \<times> sterm) list"   and x22 :: "(name, value) fmap"   and x31 :: "(name, (Term_Class.term \<times> sterm) list) fmap"   and x32 :: "name"   and x33 :: "(name, value) fmap" shows "Vabs x21 x22 \<noteq> Vrecabs x31 x32 x33"
eq': shows "(r \<star> \<r>[f]) \<cdot> \<a>[r, f, src f] \<cdot> (\<rho> \<star> src f) \<cdot> local.inv ((\<a>[local.dom \<rho>, w', w] \<cdot> (\<nu>' \<star> w) \<cdot> \<nu>) \<cdot> \<r>[g]) = (r \<star> \<theta> \<cdot> (\<theta>' \<star> w) \<cdot> \<a>\<^sup>-\<^sup>1[f, w', w]) \<cdot> \<a>[r, f, w' \<star> w] \<cdot> (\<rho> \<star> w' \<star> w)"
\<theta>: shows "\<theta> \<equiv> \<lambda>y. if Var y \<in> set U then T ! the_i y else Var y"
dropWhile_idem: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "dropWhile P (dropWhile P xs) = dropWhile P xs"
real_interval_times: fixes A :: "float interval"   and B :: "float interval" shows "real_interval (A * B) = real_interval A * real_interval B"
3: fixes n :: "String.literal"   and K :: "(String.literal \<times> String.literal set list) list"   and tcs :: "(String.literal \<times> (String.literal \<times> String.literal set list) list) list"   and c :: "String.literal"   and Ss :: "String.literal set list" assumes "(n, K) \<in> set tcs"   and "(c, Ss) \<in> set (the (lookup (\<lambda>k. k = n) tcs))"   and "exe_ars_conds tcs"   and "dom (map_of (map (apsnd map_of) tcs)) = dom (map_of tao)"   and "\<forall>type\<in>dom (map_of tao). \<forall>ars\<in>ran (the (map_of (map (apsnd map_of) tcs) type)). the (map_of tao type) = length ars" shows "the (lookup (\<lambda>k. k = n) tao) = length Ss"
swapInp_def: fixes xs :: "'varSort"   and x :: "'var"   and y :: "'var"   and inp :: "'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qTerm set option" shows "inp %[x \<and> y]_xs \<equiv> lift (swap xs x y) inp"
y: shows "y = insertion e (map_poly real_of_float p2) + ye"
rtrancl_refl: fixes a :: "'d"   and r :: "('d \<times> 'd) set" shows "(a, a) \<in> r\<^sup>*"
isCont_csqrt': fixes z :: "'a"   and f :: "'a \<Rightarrow> complex" assumes "isCont f z"   and "f z \<notin> \<real>\<^sub>\<le>\<^sub>0" shows "isCont (\<lambda>x. csqrt (f x)) z"
bdt_Some_Node_iff: fixes t :: "dag"   and var :: "ref \<Rightarrow> nat"   and bdt1 :: "bdt"   and v :: "nat"   and bdt2 :: "bdt" shows "(bdt t var = Some (Bdt_Node bdt1 v bdt2)) = (\<exists>p l r. t = Node l p r \<and> bdt l var = Some bdt1 \<and> bdt r var = Some bdt2 \<and> 1 < v \<and> var p = v)"
G: shows "set ys \<subseteq> range p \<union> range q"
ir_valid_While1: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and b :: "bexp"   and c :: "com"   and k :: "nat"   and c' :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" assumes "ir_valid (get_back P b c k) (WHILE b DO c) c' Q" shows "ir_valid (\<lambda>s s'. get_back P b c k s s' \<and> bval b s) c SKIP (\<lambda>t t'. get_back P b c (Suc k) t t' \<and> (\<exists>u u'. (WHILE b DO c, t) \<Rightarrow> u \<and> (c', t') \<Rightarrow> u' \<and> Q u u'))"
abs_le_self_iff: fixes abs :: "'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 less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "less_eq (abs a) a = less_eq zero a"
1: shows "hdomain (arr_to_hfun f) = ide_to_hf (local.dom f)"
simps_28: fixes f1 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> k \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f3 :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat \<Rightarrow> 'a"   and f4 :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and f5 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and x31 :: "nat \<Rightarrow> nat option"   and x32 :: "nat" shows "(case Rerr x31 x32 of Rreq x xa xb xc xd xe xf xg \<Rightarrow> f1 x xa xb xc xd xe xf xg | Rrep x xa xb xc xd \<Rightarrow> f2 x xa xb xc xd | Rerr x xa \<Rightarrow> f3 x xa | Newpkt x xa \<Rightarrow> f4 x xa | Pkt x xa xb \<Rightarrow> f5 x xa xb) = f3 x31 x32"
assms_2: shows "(\<exists>xr j. xr \<noteq> xs \<and> prefix xr xs \<and> j \<le> i \<and> vs @ xr \<in> RM M2 M1 \<Omega> V m j) \<or> (\<exists>xc. xc \<noteq> xs \<and> prefix xc xs \<and> vs @ xc \<in> C M2 M1 \<Omega> V m i - RM M2 M1 \<Omega> V m i)"
subst_not_dom_fixed: fixes v :: "'c"   and t :: "('d, 'c) Term.term"   and s :: "'c \<Rightarrow> ('d, 'c) Term.term" assumes "v \<in> fv t"   and "v \<notin> subst_domain s" shows "v \<in> fv (t \<cdot> s)"
mpath_weight'_correct: fixes G :: "('a, 'b) graph"   and v0 :: "'a"   and wl :: "'a \<Rightarrow> 'b infty option"   and res :: "'a \<Rightarrow> (('a \<times> 'b \<times> 'a) list \<times> 'b) option"   and v :: "'a" assumes "Dijkstra G v0"   and "Dijkstra.dinvarm (wl, res)" shows "mpath_weight' (res v) = path_weight' (Dijkstra.\<alpha>r res v)"
integral_real_density: fixes f :: "'a \<Rightarrow> real"   and M :: "'a measure"   and g :: "'a \<Rightarrow> real" assumes "f \<in> borel_measurable M"   and "AE x in M. 0 \<le> f x"   and "g \<in> borel_measurable M" shows "integral\<^sup>L (density M (\<lambda>x. ennreal (f x))) g = LINT x|M. f x * g x"
finvimageI1: fixes a :: "V"   and r :: "V"   and A :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<bullet> (r \<restriction>\<^sup>r\<^sub>\<bullet> A)" shows "a \<in>\<^sub>\<circ> r -`\<^sub>\<bullet> A"
dist_le_zero_iff: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.metric_space dist uniformity open" shows "(dist x y \<le> 0) = (x = y)"
sub_composition_closed: fixes a :: "'a"   and b :: "'a" assumes "a \<in> Units"   and "b \<in> Units" shows "a \<cdot> b \<in> Units"
degeneralize_language: fixes A :: "('c, 'd) dgca" shows "DCA.language (degeneralize A) = DGCA.language A"
prems_2: shows "n\<^sub>1 \<le> Suc n\<^sub>2"
inEx: fixes x :: "'c"   and A :: "'c set" assumes "x \<in> A" shows "\<exists>y\<in>A. y = x"
insertion_sum': fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a mpoly"   and n :: "nat" shows "insertion f (sum g {..n}) = (\<Sum>i\<le>n. insertion f (g i))"
saturatedD: fixes \<Gamma> :: "'b inference set"   and N :: "'b literal multiset set"   and CC :: "'b literal multiset multiset"   and D :: "'b literal multiset"   and E :: "'b literal multiset" assumes "inference_system.saturated \<Gamma> N"   and "Infer CC D E \<in> \<Gamma>"   and "set_mset CC \<subseteq> N"   and "D \<in> N" shows "E \<in> N"
lower_closure: shows "\<pi>\<^sup>*\<^bsub>G\<^esub> \<in> carrier \<X>\<^bsub>G\<^esub> \<rightarrow> carrier \<Y>\<^bsub>G\<^esub>"
minGraphProps_5: fixes g :: "graph"   and v :: "nat"   and f :: "face" assumes "minGraphProps g"   and "v \<in> \<V> g"   and "f \<in> set (facesAt g v)" shows "v \<in> \<V> f"
well_constrained_def: fixes C :: "'a eclause" shows "well_constrained C = (\<forall>y. y \<in> trms_ecl C \<longrightarrow> dom_trm y (cl_ecl C))"
induct: fixes P :: "'b heap list \<Rightarrow> bool"   and a0 :: "'b heap list" assumes "P []"   and "\<And>h. P [h]"   and "\<And>h1 h2 hs. P hs \<Longrightarrow> P (h1 # h2 # hs)" shows "P a0"
simps_2: fixes C :: "ty"   and X :: "ntrm set" shows "NEGc (NOT C) X = AXIOMSc (NOT C) \<union> BINDINGc (NOT C) X \<union> NOTRIGHT (NOT C) (\<parallel>(C)\<parallel>)"
equalityCE: fixes A :: "'a set"   and B :: "'a set"   and c :: "'a" assumes "A = B" obtains "c \<in> A" and "c \<in> B"   | "c \<notin> A" and "c \<notin> B"
cod_simp: fixes f :: "'a" assumes "Right.arr f" shows "Right.cod f = cod f"
distinct_9: fixes x2 :: "'a \<times> nat"   and x3 :: "'a \<times> nat" shows "Formal_in x2 \<noteq> Formal_out x3"
random_permutation_of_set: fixes A :: "'b set" assumes "finite A"   and "A \<noteq> {}" shows "pmf_of_set (permutations_of_set A) = pmf_of_set A \<bind> (\<lambda>x. pmf_of_set (permutations_of_set (A - {x})) \<bind> (\<lambda>xs. return_pmf (x # xs)))"
null_sets_density: fixes h :: "'b \<Rightarrow> ennreal"   and M :: "'b measure" assumes "h \<in> borel_measurable M"   and "AE x in M. h x \<noteq> 0" shows "null_sets (density M h) = null_sets M"
weak_gt_both_mono: fixes xys :: "('a \<times> 'a) list" assumes "\<forall>x y. (x, y) \<in> set xys \<longrightarrow> weak_gt x y" shows "\<exists>gt. SN_both_mono_ordered_semiring_1 default gt arc_pos \<and> (\<forall>x y. (x, y) \<in> set xys \<longrightarrow> gt x y)"
g_def: shows "g \<equiv> restrict1 (canonical_retraction A C \<circ> canonical_retraction B' D) (\<Union> A)"
return_envT_def: shows "Monad_Overloading.return = return_env Monad_Overloading.return"
assms_3: shows "\<forall>x. name (f x) = name x"
matrix_Tensor_elements: fixes M1 :: "'a list list"   and M2 :: "'a list list" shows "\<forall>i j. (i < row_length M1 * row_length M2 \<and> j < length M1 * length M2) \<and> mat (row_length M1) (length M1) M1 \<and> mat (row_length M2) (length M2) M2 \<longrightarrow> (M1 \<otimes> M2) ! j ! i = M1 ! (j div length M2) ! (i div row_length M2) * M2 ! (j mod length M2) ! (i mod row_length M2)"
nsmulextp_mset_code: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> bool \<times> bool"   and xs :: "'a list"   and ys :: "'a list" shows "nsmulextp f (mset xs) (mset ys) = snd (mul_ext f xs ys)"
lActive_greatest: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and t :: "nat \<Rightarrow> cnf"   and i :: "nat"   and n :: "nat" assumes "active c (t i)"   and "\<forall>n'>n. \<not> active c (t n')" shows "i \<le> dynamic_component.lActive active c t"
I2: fixes S :: "'b"   and x :: "'a"   and xs :: "'a list" assumes "invariant S"   and "is_node x"   and "list_all is_node xs"   and "memb x S \<Longrightarrow> P S xs"   and "\<not> memb x S \<Longrightarrow> P (ins x S) (succs x @ xs)" shows "P S (x # xs)"
plus_Integer_eq_bottom_iff: fixes x :: "Integer"   and y :: "Integer" shows "(x + y = \<bottom>) = (x = \<bottom> \<or> y = \<bottom>)"
simps_2: fixes x :: "'a"   and xs :: "'a list" shows "remdups (x # xs) = (if x \<in> set xs then remdups xs else x # remdups xs)"
Cons: fixes a :: "'g"   and A :: "'g set"   and l :: "'g list" assumes "a \<in> A"   and "l \<in> lists A" shows "a # l \<in> lists A"
h5: shows "invertible_mat (mat_of_rows_list 2 [[1, 1], [1, - 1]])"
onorm: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "bounded_linear f" shows "norm (f x) \<le> onorm f * norm x"
synthesized_call_def: fixes P :: "'c prog"   and h :: "'heap" shows "synthesized_call P h = (\<lambda>(a, M, vs). \<exists>T Ts Tr D. typeof_addr h a = \<lfloor>T\<rfloor> \<and> P \<turnstile> class_type_of T sees M: Ts\<rightarrow>Tr = Native in D)"
Max_le_iMin_conv_singleton: fixes I :: "'a set" assumes "finite I"   and "I \<noteq> {}" shows "(Max I \<le> iMin I) = (\<exists>x. I = {x})"
outstanding_refs_conv: fixes P :: "'b memref \<Rightarrow> bool"   and sb :: "'b memref list" shows "outstanding_refs P sb = \<Union> (address_of ` {r \<in> set sb. P r})"
thenelse: shows "eval (Cn 1 r_prod_encode [Cn 1 r_snoc [Cn 1 r_update [r_pdec1, r_pdec2, r_const 1], Z], Cn 1 r_length [r_pdec1]]) [arg] \<down>= prod_encode (e_snoc (e_update b k 1) 0, e_length b)"
cmp_inj: fixes f :: "'h \<Rightarrow> 'i"   and A :: "'h set"   and B :: "'i set"   and g :: "'i \<Rightarrow> 'j"   and C :: "'j set" assumes "f \<in> A \<rightarrow> B"   and "g \<in> B \<rightarrow> C"   and "inj_on f A"   and "inj_on g B" shows "inj_on (cmp g f) A"
fid_on_vunion: fixes A :: "V"   and B :: "V" shows "fid_on (A \<union>\<^sub>\<circ> B) = fid_on A \<union>\<^sub>\<circ> fid_on B"
bounded_linear_inner_right: fixes a :: "'c" shows "bounded_linear ((\<bullet>) a)"
single_gen_free_neg: fixes a :: "'a"   and n :: "int" assumes "sfg A a"   and "n\<triangleright>a\<^bsub>A\<^esub> = \<zero>" shows "n = 0"
strict_iff_order: fixes a :: "'a multiset"   and b :: "'a multiset" shows "(a \<subset># b) = (a \<subseteq># b \<and> a \<noteq> b)"
edge_end_tup: fixes x :: "('a \<times> 'b) \<times> 'a \<times> 'c" shows "Pre_Port_Graph.edge_end x = fst (snd x)"
col_i_rw: shows "column i (Gram_Schmidt_upt_k A (Suc k)) = column i (Gram_Schmidt_upt_k A k)"
finite_inf_closed: fixes L :: "('c, 'd) gorder_scheme"   and A :: "'c set" assumes "weak_lower_semilattice L"   and "finite A"   and "A \<subseteq> carrier L"   and "A \<noteq> {}" shows "\<Sqinter>\<^bsub>L\<^esub>A \<in> carrier L"
N_2: fixes x :: "'a" assumes "x \<notin> N" shows "f x = g x"
skip_type: fixes \<Gamma> :: "Eg1.var \<Rightarrow> bexp set option"   and \<S> :: "Eg1.var set \<times> Eg1.var set"   and P :: "bexp set" shows "has_type \<Gamma> \<S> P Skip \<Gamma> \<S> P"
NfuncPair_neq_at_def: fixes f :: "nat \<Rightarrow> 'e set"   and g :: "nat \<Rightarrow> 'e set"   and i :: "nat" shows "NfuncPair_neq_at f g i = (f i \<noteq> g i)"
c1_4: shows "D1 \<in> carrier_mat nr2 n2"
lm074: fixes XX :: "'b set set" shows "\<Union> (finestpart ` XX) \<subseteq> finestpart (\<Union> XX)"