hcju/isabelleps huggingface dataset:
swap_col_to_front_result: fixes A :: "'b mat"   and n :: "nat"   and m :: "nat"   and J :: "nat" assumes "A \<in> carrier_mat n m"   and "J < m" shows "swap_col_to_front A J = mat n m (\<lambda>(i, j). if j = 0 then A $$ (i, J) else if j \<le> J then A $$ (i, j - 1) else A $$ (i, j))"
connected_inV_iff: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and u :: "nat"   and v :: "nat" assumes "Graph.connected c u v" shows "(v \<in> Graph.V c) = (u \<in> Graph.V c)"
bij_betw_image_Fpow: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set" assumes "bij_betw f A B" shows "bij_betw ((`) f) (Fpow A) (Fpow B)"
n_ge_trans: fixes i :: "nat"   and j :: "nat" assumes "1 \<le> i"   and "i \<le> j"   and "j \<le> k" shows "n j \<le> n i"
that: fixes x :: "'b" assumes "x \<in> S" shows "norm (f x) \<le> K"
irreflexiveLexLess: fixes x :: "('b \<times> bool) list" shows "(x, x) \<notin> lexLess"
less_asym': fixes a :: "'a word"   and b :: "'a word"   and P :: "bool" assumes "a <s b"   and "b <s a" shows "P"
rtrancl_beta_AppR: fixes t :: "lam"   and t' :: "lam"   and s :: "lam" assumes "t \<rightarrow>\<^sub>\<beta>\<^sup>* t'" shows "s \<degree> t \<rightarrow>\<^sub>\<beta>\<^sup>* s \<degree> t'"
RejectIteratorTE: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and src :: "'a expr"   and k :: "call_kind"   and its :: "String.literal list"   and its_ty :: "'a OCL_Types.type option"   and body :: "'a expr"   and \<tau> :: "'a OCL_Types.type" assumes "\<Gamma> \<turnstile>\<^sub>E RejectIteratorCall src k its its_ty body : \<tau>" obtains \<sigma> :: "'a OCL_Types.type" and \<rho> :: "'a OCL_Types.type" where "k = ArrowCall" and "\<Gamma> \<turnstile>\<^sub>I (src, its, its_ty, body) : (\<tau>, \<sigma>, \<rho>)" and "length its \<le> Suc 0" and "\<rho> \<le> Boolean[?]"
twoCisTypeII_2: shows "\<forall>x\<in>cbox a b. g2 x \<le> g1 x"
lang_ENC: fixes FO :: "nat set"   and n :: "nat"   and SO :: "nat set" assumes "FO \<subseteq> {0..<n}"   and "SO \<subseteq> {0..<n} - FO" shows "lang n (ENC n FO) - {[]} = {enc (w, I) |w I. length I = n \<and> (0 < length w \<and> (\<forall>a\<in>set w. a \<in> set \<Sigma>) \<and> (\<forall>x\<in>set I. case x of Inl p \<Rightarrow> p < length w | Inr P \<Rightarrow> \<forall>p\<in>P. p < length w)) \<and> (\<forall>i\<in>FO. case I ! i of Inl x \<Rightarrow> True | Inr x \<Rightarrow> False) \<and> (\<forall>i\<in>SO. case I ! i of Inl x \<Rightarrow> False | Inr x \<Rightarrow> True)}"
elims_1: fixes x :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool"   and xa :: "'b literal set set"   and y :: "bool" assumes "validate_clause_set x xa = y" obtains I :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool" and S :: "'b literal set set" where "x = I" and "xa = S" and "y = (\<forall>C. C \<in> S \<longrightarrow> validate_clause I C)"
simps_3: fixes x21 :: "'f"   and x22 :: "'f list" shows "x21 # x22 \<noteq> []"
semilattice_neutr_axioms: shows "semilattice_neutr (\<squnion>) \<bottom>"
notin_set_nthsI: fixes x :: "'c"   and xs :: "'c list"   and I :: "nat set" assumes "x \<notin> set xs" shows "x \<notin> set (nths xs I)"
actions_ok1_ex_actions_ok2: fixes s1 :: "('d, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'd \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'e wait_set_status option) \<times> 't set"   and t :: "'t"   and ta1 :: "('d, 't, 'x1, 'm1, 'e, 'f) thread_action"   and ta2 :: "('d, 't, 'x2, 'm2, 'e, 'f) thread_action" assumes "r1.actions_ok s1 t ta1"   and "ta1 \<sim>m ta2" obtains s2 :: "('d, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'd \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'e wait_set_status option) \<times> 't set" where "r2.actions_ok s2 t ta2"
ramanujan_sum_k_n_dirichlet_expr: fixes k :: "nat"   and n :: "nat" assumes "0 < k"   and "0 < n" shows "ramanujan_sum k n = of_nat (totient k) * moebius_mu (k div gcd n k) / of_nat (totient (k div gcd n k))"
assms_1: shows "s O ns \<subseteq> (s \<union> ns)\<^sup>* O s"
homotopic_loops: fixes s :: "'b set"   and p :: "real \<Rightarrow> 'b"   and q :: "real \<Rightarrow> 'b" shows "homotopic_loops s p q = (\<exists>h. continuous_on ({0..1} \<times> {0..1}) h \<and> h ` ({0..1} \<times> {0..1}) \<subseteq> s \<and> (\<forall>x\<in>{0..1}. h (0, x) = p x) \<and> (\<forall>x\<in>{0..1}. h (1, x) = q x) \<and> (\<forall>t\<in>{0..1}. pathfinish (h \<circ> Pair t) = pathstart (h \<circ> Pair t)))"
measurable_embed_measure2: fixes f :: "'e \<Rightarrow> 'f"   and M :: "'e measure" assumes "inj f" shows "f \<in> M \<rightarrow>\<^sub>M embed_measure M f"
StaticUpDynCast: fixes 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"   and E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and a :: "nat"   and Cs :: "char list list"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and C :: "char list"   and Cs' :: "char list list"   and Ds :: "char list list" assumes "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>ref (a, Cs),s\<^sub>1\<rangle>"   and "P \<turnstile> Path last Cs to C unique"   and "P \<turnstile> Path last Cs to C via Cs' "   and "Ds = Cs @\<^sub>p Cs'" shows "P,E \<turnstile> \<langle>Cast C e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>ref (a, Ds),s\<^sub>1\<rangle>"
eq1: shows "\<phi> ((6, 2) [^]\<^bsub>R\<^esub> M) = \<phi> (6, - 2)"
of_int_diff: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and w :: "int"   and z :: "int" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus (w - z) = minus (ring_1.of_int one plus zero minus w) (ring_1.of_int one plus zero minus z)"
map_\<I>_full: fixes f :: "'h \<Rightarrow> 'j"   and g :: "'k \<Rightarrow> 'i" shows "map_\<I> f g \<I>_full = \<I>_uniform UNIV (range g)"
hyps: fixes x2aa :: "('sf, 'sz) trm" assumes "x2aa \<in> range x2a"   and "Vagree \<nu> \<omega> (\<Union>i\<in>SIGT x2aa. case SFunctions \<sigma> i of None \<Rightarrow> {} | Some a \<Rightarrow> FVT a)" shows "sterm_sem (local.adjoint I \<sigma> \<nu>) x2aa = sterm_sem (local.adjoint I \<sigma> \<omega>) x2aa"
ass: shows "nat_to_bv (bv_to_nat w) = norm_unsigned w"
terminal_TCons: fixes x :: "'b"   and xs :: "('b, 'a) tllist" shows "terminal (TCons x xs) = terminal xs"
list_of_oalist_ntm_inject: fixes x :: "('a, 'b) oalist_ntm"   and y :: "('a, 'b) oalist_ntm" shows "(list_of_oalist_ntm x = list_of_oalist_ntm y) = (x = y)"
groupF_Union_set: fixes f :: "'c \<Rightarrow> 'd"   and xs :: "'c list" shows "\<Union> (set ` set (groupF f xs)) = set xs"
llength_eq_0: fixes xs :: "'b llist" shows "(llength xs = 0) = lnull xs"
wf_design: shows "design \<V>\<^sup>+ \<B>\<^sup>+"
elims_2: fixes G :: "('a, 'b) graph"   and x :: "'a"   and xa :: "'a list" assumes "valid_unSimpGraph G"   and "valid_unSimpGraph.adj_path G x xa" obtains v :: "'a" where "x = v__" and "xa = []" and "v__ \<in> nodes G"   | v :: "'a" and u :: "'a" and us :: "'a list" where "x = v" and "xa = u # us" and "valid_unMultigraph.adjacent G v u \<and> valid_unSimpGraph.adj_path G u us"
eq_add_str: fixes U :: "'b set" assumes "U \<subseteq> X" shows "ind_add_str U = add_str U"
shSubobj_isSubobj: fixes 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"   and C :: "char list"   and Xs :: "char list list"   and X :: "char list"   and Y :: "char list" assumes "is_subobj P (C, Xs @ [X])"   and "P \<turnstile> X \<prec>\<^sub>S Y" shows "is_subobj P (C, [Y])"
fbrelation_axioms: shows "fbrelation (\<D>\<^sub>\<circ> f)"
inv_finals_nonempty: fixes g :: "graph" assumes "Invariants.inv g" shows "finals g \<noteq> []"
countable_Image: fixes Y :: "'c set"   and X :: "('c \<times> 'd) set" assumes "\<And>y. y \<in> Y \<Longrightarrow> countable (X `` {y})"   and "countable Y" shows "countable (X `` Y)"
estep_Pr: fixes c :: "(recf \<times> nat list \<times> nat list) list \<times> nat option"   and n :: "nat"   and f :: "recf"   and g :: "recf"   and xs :: "nat list"   and ls :: "nat list"   and fs :: "(recf \<times> nat list \<times> nat list) list"   and rv :: "nat option" assumes "c = ((Pr n f g, xs, ls) # fs, rv)"   and "recfn (length xs) (Pr n f g)" shows "estep_Pr (encode_config c) = encode_config (step c)"
single_one: shows "monomial (1::'c) (0::'b) = 1"
risky_asset_martingale_only_if: fixes N :: "bool stream measure"   and q :: "real" assumes "N = bernoulli_stream q"   and "0 < q"   and "q < 1"   and "martingale N G (discounted_value r (prices Mkt stk))" shows "q = (1 + r - d) / (u - d)"
minor: fixes y :: "V" assumes "y < x" shows "y + x = x"
pelims: fixes x :: "'d"   and xa :: "('c, 'e) step"   and y :: "('c, 'd \<times> 'e) step" assumes "fix_step x xa = y"   and "fix_step_dom (x, xa)" obtains a :: "'d" where "x = a__" and "xa = Done" and "y = Done" and "fix_step_dom (a__, Done)"   | a :: "'d" and s :: "'e" where "x = aa__" and "xa = Skip sa__" and "y = Skip (aa__, sa__)" and "fix_step_dom (aa__, Skip sa__)"   | a :: "'d" and xb :: "'c" and s :: "'e" where "x = a" and "xa = Yield xb s" and "y = Yield xb (a, s)" and "fix_step_dom (a, Yield xb s)"
linordered_field_no_lb: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "\<forall>x. \<exists>y. less y x"
j_OUT_eq: fixes x :: "'v" assumes "x \<in> A \<Gamma>" shows "(\<Sum>\<^sup>+ y\<in>\<^bold>O\<^bold>U\<^bold>T x. j (\<langle>x\<rangle>, \<langle>y\<rangle>)) = d_OUT j \<langle>x\<rangle>"
pull_def: fixes x :: "'d \<times> 'e \<times> 'f" shows "pull x \<equiv> ((fst x, fst (snd x)), snd (snd x))"
VectorSpaceHom_idhom: shows "VectorSpaceHom (\<cdot>) V (\<cdot>) (id \<down> V)"
card_eq_dim: fixes B :: "'c set"   and V :: "'c set" assumes "B \<subseteq> V"   and "card B = vs2.dim V"   and "finite B" shows "vs2.independent B = (V \<subseteq> vs2.span B)"
impl_sup_bl: fixes a :: "'a"   and b :: "'a" shows "a l\<rightarrow> b \<squnion> (b l\<rightarrow> a) = (1::'a)"
subsumes_reflexive: fixes t :: "transition"   and c :: "nat \<Rightarrow>f value option" shows "subsumes t c t"
eq_iff: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> A" shows "(x = y) = (x \<sqsubseteq> y \<and> y \<sqsubseteq> x)"
subsetD: fixes A :: "'d set"   and B :: "'d set"   and c :: "'d" assumes "A \<subseteq> B"   and "c \<in> A" shows "c \<in> B"
permutes_bij': fixes A :: "'b set"   and a_to_b :: "'b \<Rightarrow> 'c"   and B :: "'c set"   and b_to_a :: "'c \<Rightarrow> 'b" assumes "\<And>a. a \<in> A \<Longrightarrow> a_to_b a \<in> B"   and "\<And>b. b \<in> B \<Longrightarrow> b_to_a b \<in> A"   and "\<And>a. a \<in> A \<Longrightarrow> b_to_a (a_to_b a) = a"   and "\<And>b. b \<in> B \<Longrightarrow> a_to_b (b_to_a b) = b" shows "{p. p permutes A} = (\<lambda>p a. if a \<in> A then b_to_a (p (a_to_b a)) else a) ` {p. p permutes B}"
subgame_VV: fixes G :: "('c, 'd) ParityGame_scheme"   and p :: "Player"   and V' :: "'c set" assumes "ParityGame G" shows "(if p = Even then V0\<^bsub>ParityGame.subgame G V'\<^esub> else V\<^bsub>ParityGame.subgame G V'\<^esub> - V0\<^bsub>ParityGame.subgame G V'\<^esub>) = V' \<inter> (if p = Even then V0\<^bsub>G\<^esub> else V\<^bsub>G\<^esub> - V0\<^bsub>G\<^esub>)"
empty_foldSetDE: fixes x :: "'c"   and D :: "'c set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and e :: "'c" assumes "({}, x) \<in> foldSetD D f e" obtains "x = e" and "e \<in> D"
d_def: shows "d = (\<lambda>x. (f x + f (x + 1)) /\<^sub>R 2 - integral {x..x + 1} f)"
Pure: fixes x :: "'h"   and gpv :: "('h, 'i, 'j) gpv"   and \<Gamma> :: "('i, 'j) \<I>" assumes "Pure x \<in> set_spmf (the_gpv gpv)" shows "x \<in> results_gpv \<Gamma> gpv"
subadditive_def: fixes u :: "nat \<Rightarrow> real" shows "subadditive u = (\<forall>m n. u (m + n) \<le> u m + u n)"
Id_on_iff: fixes x :: "'b"   and y :: "'b"   and A :: "'b set" shows "((x, y) \<in> s2r A) = (x = y \<and> x \<in> A)"
distinct_25: fixes x4 :: "'b" shows "WakeUp x4 \<noteq> Notified"
len_decomp: fixes xs :: "nat list"   and ys :: "nat list"   and y :: "nat"   and zs :: "nat list"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and x :: "nat"   and z :: "nat" assumes "xs = ys @ y # zs" shows "len m x z xs = len m x y ys + len m y z zs"
subtermeq_imp_subtermtypeeq: fixes t :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term"   and s :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term" assumes "wf\<^sub>t\<^sub>r\<^sub>m t"   and "s \<sqsubseteq> t" shows "\<Gamma> s \<sqsubseteq> \<Gamma> t"
arr_dom: fixes f :: "('o, 'a) arr" assumes "arr f" shows "arr (local.dom f)"
less_imp_inverse_less_neg: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less a b"   and "less b zero" shows "less (inverse b) (inverse a)"
mf_id: shows "mf q (A $$ (k, l)) = (q', p', g)"
StateParallel_frame: fixes a :: "'d \<Longrightarrow> 'e"   and b :: "'f \<Longrightarrow> 'e"   and P :: "'e hrel"   and Q :: "'e hrel" assumes "vwb_lens a"   and "vwb_lens b"   and "a \<bowtie> b" shows "{&a, &b}:[P |a|b|\<^sub>\<sigma> Q] = P |a|b|\<^sub>\<sigma> Q"
obtain_sum_ideal_generated: 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 a :: "'b"   and A :: "'b set" assumes "class.comm_ring_1 times one plus zero minus uminus"   and "a \<in> ring.ideal_generated plus zero uminus times A"   and "finite A" obtains f :: "'b \<Rightarrow> 'b" where "comm_monoid_add.sum plus zero (\<lambda>i. times (f i) i) A = a"
det_cofactor: shows "Determinant.cofactor (mat_delete (make_mat n k B) 0 k) (k - 1) 0 = (- (1::'a)) ^ (k - 1)"
lex_ext_unbounded_map: fixes ss :: "'a list"   and ts :: "'a list"   and r :: "'a \<Rightarrow> 'a \<Rightarrow> bool \<times> bool"   and f :: "'a \<Rightarrow> 'a" assumes "\<And>i. \<lbrakk>i < length ss; i < length ts; fst (r (ss ! i) (ts ! i))\<rbrakk> \<Longrightarrow> fst (r (map f ss ! i) (map f ts ! i))"   and "\<And>i. \<lbrakk>i < length ss; i < length ts; snd (r (ss ! i) (ts ! i))\<rbrakk> \<Longrightarrow> snd (r (map f ss ! i) (map f ts ! i))" shows "(fst (lex_ext_unbounded r ss ts) \<longrightarrow> fst (lex_ext_unbounded r (map f ss) (map f ts))) \<and> (snd (lex_ext_unbounded r ss ts) \<longrightarrow> snd (lex_ext_unbounded r (map f ss) (map f ts)))"
ground_fmla_aux_simps_2: fixes t :: "trm"   and u :: "trm"   and S :: "atom set" shows "ground_fmla_aux (t EQ u) S = (ground_aux t S \<and> ground_aux u S)"
well_base_filter: fixes base :: "nat"   and M :: "'a multiset"   and p :: "'a \<Rightarrow> bool" assumes "2 \<le> base"   and "well_base base M" shows "well_base base (filter_mset p M)"
prems_5: shows "p = hd_prog p\<^sub>s\<^sub>b ((r # sb) @ sb')"
reaches1_reaches_iff1: fixes E :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" shows "E\<^sup>+\<^sup>+ x y = (\<exists>z. E x z \<and> E\<^sup>*\<^sup>* z y)"
polynomial_in_carrier: fixes p :: "'a list" assumes "polynomial K p" shows "set p \<subseteq> carrier R"
inj_on_xor_list_nlists: fixes n :: "nat"   and xs :: "'a list" assumes "n \<le> length xs" shows "inj_on (([\<oplus>]) xs) (nlists UNIV n)"
rel_compp_Grp: fixes R1 :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_sum R1 R2 = (BNF_Def.Grp {x. Basic_BNFs.setl x \<subseteq> {(x, y). R1 x y} \<and> Basic_BNFs.setr x \<subseteq> {(x, y). R2 x y}} (map_sum fst fst))\<inverse>\<inverse> OO BNF_Def.Grp {x. Basic_BNFs.setl x \<subseteq> {(x, y). R1 x y} \<and> Basic_BNFs.setr x \<subseteq> {(x, y). R2 x y}} (map_sum snd snd)"
sectionE: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "category C"   and "category.section C f" obtains g :: "'b" where "partial_magma.ide C (C g f)"
replicate_mset_eq_iff: fixes m :: "nat"   and a :: "'k"   and n :: "nat"   and b :: "'k" shows "(replicate_mset m a = replicate_mset n b) = (m = 0 \<and> n = 0 \<or> m = n \<and> a = b)"
length_mvbox: fixes m :: "nat"   and n :: "nat" shows "length (mv_box m n) = 3"
noLowDPs_def: fixes S :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list" shows "noLowDPs S = (case S of (s, h) \<Rightarrow> (\<forall>x l. CONTEXT x = low \<longrightarrow> s x = RVal (Loc l) \<longrightarrow> l \<in> Dom h) \<and> (\<forall>ll c F f l. lookup h ll = Some (c, F) \<longrightarrow> GAMMA f = low \<longrightarrow> lookup F f = Some (RVal (Loc l)) \<longrightarrow> l \<in> Dom h))"
prems: shows "range_vars \<delta> \<inter> bvars\<^sub>s\<^sub>s\<^sub>t (s # S) = {}"
scheme_substitutions_only_on_free_variables: fixes sch :: "type_scheme"   and S :: "nat \<Rightarrow> typ"   and S' :: "nat \<Rightarrow> typ" assumes "\<forall>x\<in>free_tv sch. S x = S' x" shows "$ S sch = $ S' sch"
simps_9: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a" shows "(case GT of LT \<Rightarrow> f1 | EQ \<Rightarrow> f2 | GT \<Rightarrow> f3) = f3"
asymmetric_irreflexive: fixes x :: "'a" assumes "asymmetric x" shows "irreflexive x"
tendsto_rabs_zero_iff: fixes f :: "'a \<Rightarrow> real"   and F :: "'a filter" shows "((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> 0) F = (f \<longlongrightarrow> 0) F"
vsv_vimageI2: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (\<CC>\<lparr>Dom\<rparr>)" shows "\<CC>\<lparr>Dom\<rparr>\<lparr>a\<rparr> \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> (\<CC>\<lparr>Dom\<rparr>)"
n'_2: shows "indexed_eval_aux (Qs @ [Q]) i n' \<noteq> \<zero>"
return_val_def: fixes x :: "val" shows "return_val x = return (stock_measure (val_type x)) x"
seqI: fixes f :: "'a"   and g :: "'a" assumes "A.arr f"   and "A.arr g"   and "A.dom g = A.cod f" shows "A.seq g f"
density_unique: fixes f :: "'a \<Rightarrow> ennreal"   and f' :: "'a \<Rightarrow> ennreal" assumes "f \<in> borel_measurable M"   and "f' \<in> borel_measurable M"   and "density M f = density M f'" shows "AE x in M. f x = f' x"
vwb_lens_wb: fixes x :: "'f \<Longrightarrow> 'g" assumes "vwb_lens x" shows "wb_lens x"
even_odd_interpolate_def: fixes u :: "nat \<Rightarrow> 'c"   and v :: "nat \<Rightarrow> 'c"   and n :: "nat" shows "even_odd_interpolate u v n = (if even n then u (n div 2) else v (n div 2))"
Lambert_W_asymp_equiv_at_top: shows "(\<lambda>x. Lambert_W x - ln x) \<sim>[at_top] (\<lambda>x. - ln (ln x))"
cas_takeI: fixes u :: "'a"   and p :: "'b list"   and v :: "'a"   and n :: "nat"   and v' :: "'a" assumes "cas u p v"   and "awlast u (take n p) = v'" shows "cas u (take n p) v'"
finite_chain_def: fixes below :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and C :: "nat \<Rightarrow> 'd" assumes "class.po below" shows "po.finite_chain below C = (po.chain below C \<and> (\<exists>i. po.max_in_chain i C))"
choice_iff: fixes Q :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "(\<forall>x. \<exists>y. Q x y) = (\<exists>f. \<forall>x. Q x (f x))"
K1_def: shows "K1 = map_of (map (\<lambda>(s, \<mu>). (s, map_of \<mu>)) K)"
assms_2: shows "CT;\<Gamma> \<turnstile> e : C"
assms_1: shows "\<forall>y\<in>A. B y \<in> sigma_sets X Y"
f1: shows "\<forall>n r. truncate_up n r + truncate_down n (- 1 * r) = 0"
r_def: shows "r = run\<^sub>t \<delta>\<^sub>\<R> q\<^sub>\<R> w"
rtrancl_NS: fixes S :: "('a \<times> 'a) set"   and NS :: "('a \<times> 'a) set" assumes "pre_order_pair S NS" shows "NS\<^sup>* = NS"
second_vertices_def: shows "G.second_vertices \<equiv> G.second_vertex ` paths"
finite_verts: fixes G :: "('c, 'd) pre_digraph" assumes "fin_digraph G" shows "finite (verts G)"
SeqWRP_graph_aux_def: shows "SeqWRP_graph_aux \<equiv> lfp (\<lambda>p x1 x2. \<exists>SeqWRP_aux l s k sl y. x1 = (s, k, y) \<and> x2 = LstSeqP s k y AND HPair Zero Zero IN s AND All2 l k (SyntaxN.Ex sl (HPair (Var l) (Var sl) IN s AND HPair (SUCC (Var l)) (Q_Succ (Var sl)) IN s)) \<and> atom l \<sharp> (s, k, sl) \<and> atom sl \<sharp> s)"
borel_measurable_piecewise_mono: fixes C :: "real set set"   and f :: "real \<Rightarrow> real" assumes "countable C"   and "\<And>c. c \<in> C \<Longrightarrow> c \<in> sets borel"   and "\<And>c. c \<in> C \<Longrightarrow> mono_on f c"   and "\<Union> C = UNIV" shows "f \<in> borel_measurable borel"
siso_Sbis: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "('b, 'c) com" assumes "PL_Indis indis"   and "PL_Indis.siso tval aval indis c" shows "(c, c) \<in> PL_Indis.Sbis tval aval indis"
name_fresh_abs: fixes b :: "name"   and a :: "name"   and x :: "'a" shows "b \<sharp> [a].x = (b = a \<or> b \<sharp> x)"
matrix_compose_gen: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec"   and g :: "('a, 'c) vec \<Rightarrow> ('a, 'd) vec" assumes "linear (*s) (*s) f"   and "linear (*s) (*s) g" shows "matrix (g \<circ> f) = matrix g ** matrix f"
map_closed: fixes a :: "'a" assumes "a \<in> A" shows "f a \<in> B"
add_less_cancel_right: fixes a :: "'a"   and c :: "'a"   and b :: "'a" shows "(a + c < b + c) = (a < b)"
Restrp_tranclp_well_founded: shows "well_founded UNIV ((\<sqsubset>) \<restriction> A)\<^sup>+\<^sup>+"
sm_DF: fixes k :: "nat" shows "strict_mono_sets UNIV (DF k)"
I_def: shows "I = integral {a..b} (\<lambda>t. sum_upto \<chi> t * complex_of_real (f' t))"
ordertype_WW_1: shows "ordertype WW (lenlex less_than) \<le> ordertype UNIV (lenlex less_than)"
distr: fixes B :: "'b set" assumes "B \<in> sets M'" shows "emeasure M (X -` B \<inter> space M) = emeasure M' (A \<inter> B) / emeasure M' A"
conditional_mutual_information_def: fixes b :: "real"   and MX :: "'b measure"   and MY :: "'f measure"   and MZ :: "'g measure"   and X :: "'c \<times> 'd \<times> 'e \<Rightarrow> 'b"   and Y :: "'c \<times> 'd \<times> 'e \<Rightarrow> 'f"   and Z :: "'c \<times> 'd \<times> 'e \<Rightarrow> 'g" shows "P.conditional_mutual_information b MX MY MZ X Y Z \<equiv> P.mutual_information b MX (MY \<Otimes>\<^sub>M MZ) X (\<lambda>x. (Y x, Z x)) - P.mutual_information b MX MZ X Z"
add_sum: fixes i :: "nat"   and j :: "nat"   and k :: "nat" assumes "0 < i"   and "i \<le> j"   and "j < k" shows "local.sum i j + local.sum (Suc j) k = local.sum i k"
exhaust: fixes y :: "'b hp" obtains x1 :: "'b" and x2 :: "'b hp list" where "y = Hp x1 x2"
bigo_mult2: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" shows "f *o O(g) \<subseteq> O(f * g)"