hcju/isabelleps huggingface dataset:
in_set_least_squares_approximation_eq_full_rank_QR: fixes A :: "((real, 'a) vec, 'b) vec"   and x :: "(real, 'a) vec"   and b :: "(real, 'b) vec" assumes "rank A = ncols A" shows "(x \<in> set_least_squares_approximation A b) = (snd (QR_decomposition A) *v x = Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)"
decrpoly: fixes t :: "poly"   and x :: "'a"   and bs :: "'a list" assumes "polybound0 t" shows "\<lparr>t\<rparr>\<^sub>p\<^bsup>x # bs\<^esup> = \<lparr>decrpoly t\<rparr>\<^sub>p\<^bsup>bs\<^esup>"
agents_eq: shows "agents = {i. permute_profile \<sigma> R i \<noteq> (\<lambda>_ _. False)}"
parallel_fix_ind: fixes P :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and F :: "'r \<rightarrow> 'r"   and G :: "'s \<rightarrow> 's" assumes "adm (\<lambda>x. P (prod.fst x) (prod.snd x))"   and "P \<bottom> \<bottom>"   and "\<And>x y. P x y \<Longrightarrow> P (F\<cdot>x) (G\<cdot>y)" shows "P (fix\<cdot>F) (fix\<cdot>G)"
evaln_elim_cases_20: fixes G :: "prog"   and s :: "st"   and e :: "expr"   and c1 :: "stmt"   and c2 :: "stmt"   and n :: "nat"   and x :: "vals"   and s' :: "state" assumes "G\<turnstile>Norm s \<midarrow>In1r (If(e) c1 Else c2)\<succ>\<midarrow>n\<rightarrow> (x, s')" obtains b :: "val" and s1 :: "state" where "x = \<diamondsuit>" and "G\<turnstile>Norm s \<midarrow>e-\<succ>b\<midarrow>n\<rightarrow> s1" and "G\<turnstile>s1 \<midarrow>(if the_Bool b then c1 else c2)\<midarrow>n\<rightarrow> s'"
elims_3: fixes x :: "checker_type"   and xa :: "t list" assumes "\<not> c_types_agree x xa" obtains ts :: "t list" and ts' :: "t list" where "x = Type ts__" and "xa = ts'__" and "ts__ \<noteq> ts'__"   | ts :: "ct list" and ts' :: "t list" where "x = TopType ts" and "xa = ts'" and "\<not> ct_suffix ts (to_ct_list ts')"   | uu_ :: "t list" where "x = Bot" and "xa = uu_"
of_hypnat_add: fixes m :: "nat star"   and n :: "nat star" shows "of_hypnat (m + n) = of_hypnat m + of_hypnat n"
pos': fixes x' :: "'a" assumes "x' \<in> X"   and "dist x' x < d" shows "0 < g x'"
cspan_butterfly_UNIV: fixes basisA :: "'a set"   and basisB :: "'b set" assumes "cspan basisA = UNIV"   and "cspan basisB = UNIV"   and "is_ortho_set basisB"   and "\<And>b. b \<in> basisB \<Longrightarrow> norm b = 1" shows "cspan {butterfly a b |a b. a \<in> basisA \<and> b \<in> basisB} = UNIV"
targetsD_2: fixes b :: "'a"   and \<mu> :: "'a" assumes "b \<in> targets \<mu>" shows "weak_unit b"
H: shows "\<Gamma> \<turnstile> (\<lambda><:S. s) : (\<forall><:U. U')"
BPd'_BCh'_mechanism_domain: shows "ContractsWithBilateralSubstitutesAndIRC Xd Xh BPd' BCh'"
measurable_subprob_algebra: fixes M :: "'c measure"   and K :: "'c \<Rightarrow> 'd measure"   and N :: "'d measure" assumes "\<And>a. a \<in> space M \<Longrightarrow> subprob_space (K a)"   and "\<And>a. a \<in> space M \<Longrightarrow> sets (K a) = sets N"   and "\<And>A. A \<in> sets N \<Longrightarrow> (\<lambda>a. emeasure (K a) A) \<in> borel_measurable M" shows "K \<in> M \<rightarrow>\<^sub>M subprob_algebra N"
interior_of_eq_empty: fixes X :: "'c topology"   and S :: "'c set" shows "(X interior_of S = {}) = (\<forall>T. openin X T \<and> T \<subseteq> S \<longrightarrow> T = {})"
assms: shows "\<forall>x\<in>vars p. (0 < coeff p x \<longrightarrow> v x \<le> (0::'a)) \<and> (coeff p x < 0 \<longrightarrow> (0::'a) \<le> v x)"
C_not_Liminf: shows "C \<notin> Liminf_llist Ns"
less_mask: fixes n :: "nat" assumes "Suc 0 < n" shows "n < mask n"
assoc_simps\<^sub>A\<^sub>W\<^sub>C_2: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'c"   and g :: "'c"   and h :: "'c" assumes "associative_weak_composition V H \<a>"   and "partial_magma.ide V f"   and "partial_magma.ide V g"   and "partial_magma.ide V h"   and "H f g \<noteq> partial_magma.null V"   and "H g h \<noteq> partial_magma.null V" shows "partial_magma.dom V (\<a> f g h) = H (H f g) h"
simps_5: fixes v :: "('e, 'f, 'g) Tree"   and va :: "('e, 'f, 'g) Tree set['e]" shows "(tNot v =\<^sub>\<alpha> tConj va) = False"
dg_prod_2_ObjI': fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and ab :: "V"   and a :: "V"   and b :: "V" assumes "digraph \<alpha> \<AA>"   and "digraph \<alpha> \<BB>"   and "ab = [a, b]\<^sub>\<circ>"   and "a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>"   and "b \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>" shows "ab \<in>\<^sub>\<circ> (\<AA> \<times>\<^sub>D\<^sub>G \<BB>)\<lparr>Obj\<rparr>"
smallomega: fixes \<epsilon> :: "real" assumes "0 < \<epsilon>" shows "g \<in> \<omega>(\<lambda>x. exp ((1 - \<epsilon>) * x))"
collect_interruptsE: fixes t' :: "'b"   and ias' :: "'b interrupt_action list" assumes "t' \<in> collect_interrupts ias'" obtains n' :: "nat" where "n' < length ias'" and "ias' ! n' = IsInterrupted t' True" and "Interrupt t' \<notin> set (take n' ias')"
"3_1": fixes x :: "'d" assumes "comp kx ky = Eq"   and "x = f kx vx vy"   and "oalist_inv_raw xs"   and "oalist_inv_raw ys"   and "map2_val_compat g"   and "map2_val_compat h" shows "fst ` set (map2_val_pair f g h xs ys) \<subseteq> fst ` set xs \<union> fst ` set ys"
preserves_cod: fixes f :: "'c" assumes "C.arr f" shows "D.cod (F f) = F (C.cod f)"
el: shows "(S::'a formula set) \<in> (C::'a formula set set)"
infnorm_def: fixes x :: "'b" shows "infnorm x = Sup {\<bar>x \<bullet> b\<bar> |b. b \<in> Basis}"
ord_pp_def: fixes cmp_term :: "'a nat_term_order"   and s :: "'a"   and t :: "'a" shows "ord_pp_punit cmp_term s t = le_of_nat_term_order cmp_term (fst (s, the_min)) (fst (t, the_min))"
tau_mpj_bilin4: fixes M :: "('e, 'a, 'f) Module_scheme"   and N :: "('g, 'a, 'f) Module_scheme"   and P :: "('e \<times> 'g, 'a, 'f) Module_scheme"   and m :: "'e"   and n :: "'g"   and a :: "'a" assumes "R module M"   and "R module N"   and "FM\<^bsub>R\<^esub> P M N"   and "m \<in> carrier M"   and "n \<in> carrier N"   and "a \<in> carrier R" shows "mpj P (TR\<^bsub>R\<^esub> M N P) (m, a \<cdot>\<^sub>s\<^bsub>N\<^esub> n) = a \<cdot>\<^sub>s\<^bsub>M \<^bsub>P\<^esub>\<Otimes>\<^bsub>R\<^esub> N\<^esub> mpj P (TR\<^bsub>R\<^esub> M N P) (m, n)"
call_return_edges: fixes a :: "'edge"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and r :: "'ret"   and p :: "'pname"   and fs :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list"   and a' :: "'edge" assumes "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"   and "a' \<in> get_return_edges a" shows "\<exists>Q' f'. kind a' = Q'\<hookleftarrow>\<^bsub>p\<^esub>f'"
tl_rev: fixes p :: "'b list" shows "tl (rev p) = rev (butlast p)"
simps_4: fixes x2 :: "String.literal"   and x1 :: "int" shows "value.Str x2 \<noteq> Num x1"
dom_cod: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a" assumes "category C"   and "partial_magma.arr C f" shows "partial_magma.dom C (partial_magma.cod C f) = partial_magma.cod C f"
prod_principal_ideal: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R" shows "R \<diamondsuit>\<^sub>p a \<diamondsuit>\<^sub>r R \<diamondsuit>\<^sub>p b = R \<diamondsuit>\<^sub>p (a \<cdot>\<^sub>r b)"
measurable_count_space_insert: fixes s :: "'a"   and S :: "'a set"   and A :: "'a set" assumes "s \<in> S"   and "A \<in> sets (count_space S)" shows "insert s A \<in> sets (count_space S)"
abs_ast_operator_section_valid: shows "list_all (is_valid_operator_sas_plus abs_prob) abs_ast_operator_section"
cobounded2: fixes b :: "'c DBMEntry"   and a :: "'c DBMEntry" shows "b \<preceq> linordered_monoid.max a b"
fundantivertex_bij_betw: shows "bij_betw fundantivertex (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}) C0"
rw2: shows "fst (snd (foldl echelon_form_of_column_k_det_iarrays (1::'a, matrix_to_iarray A, 0, bezout) [0..<Suc k])) = matrix_to_iarray (fst (snd (foldl (echelon_form_of_column_k_det bezout) (1::'a, A, 0) [0..<Suc k])))"
rep_ltln\<^sub>Q_def: shows "rep_ltln\<^sub>Q \<equiv> quot_type.rep Rep_ltln\<^sub>Q"
coplanar_5: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and \<alpha> :: "Points"   and \<beta> :: "Points" assumes "rk {A, B, C} = 3"   and "rk {B, C, \<alpha>} = 2"   and "rk {A, C, \<beta>} = 2" shows "rk {A, B, C, \<alpha>, \<beta>} = 3"
length_Suc_rev_conv: fixes xs :: "'b list"   and n :: "nat" shows "(length xs = Suc n) = (\<exists>ys y. xs = ys @ [y] \<and> length ys = n)"
split_list_prop: fixes xs :: "'g list"   and P :: "'g \<Rightarrow> bool" assumes "\<exists>x\<in>set xs. P x" shows "\<exists>ys x zs. xs = ys @ x # zs \<and> P x"
set_attribute_locs_def: shows "set_attribute_locs \<equiv> l_set_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_attribute_locs"
enabledPrefixSingle: fixes SES :: "('c, 'd) SES_rec"   and s :: "'c"   and t :: "'d list"   and e :: "'d" assumes "enabled SES s (t @ [e])" shows "enabled SES s t"
eval_embed_true: fixes P :: "'a \<Rightarrow> bool"   and s :: "'a" assumes "P s" shows "\<guillemotleft> P \<guillemotright> s = 1"
elims: fixes x :: "'at Literal set"   and xa :: "'at Literal set"   and xb :: "'at"   and y :: "'at Literal set" assumes "resolvent_upon x xa xb = y" obtains P1 :: "'at Literal set" and P2 :: "'at Literal set" and A :: "'at" where "x = P1" and "xa = P2" and "xb = A" and "y = P1 - {Pos A} \<union> (P2 - {Neg A})"
insert': fixes I :: "'b set"   and p :: "'b \<Rightarrow> 'c"   and i :: "'b" assumes "finite {x \<in> I. p x \<noteq> (0::'c)}" shows "sum' p (insert i I) = (if i \<in> I then sum' p I else p i + sum' p I)"
prodM_def: fixes R :: "('j, 'k) Ring_scheme"   and I :: "'h set"   and A :: "'h \<Rightarrow> ('i, 'j, 'l) Module_scheme" shows "m\<Pi>\<^bsub>R I\<^esub> A = \<lparr>carrier = carr_prodag I A, pop = prod_pOp I A, mop = prod_mOp I A, zero = prod_zero I A, sprod = prodM_sprod R I A\<rparr>"
equals0D: fixes A :: "'e set"   and a :: "'e" assumes "A = {}" shows "a \<notin> A"
prems_2: shows "wellformed_policy1 (ys @ [y])"
length_permute_list: fixes f :: "nat \<Rightarrow> nat"   and xs :: "'b list" shows "length (permute_list f xs) = length xs"
call_zero_iff: fixes scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and scaleC :: "complex \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and cinner :: "'c \<Rightarrow> 'c \<Rightarrow> complex"   and x :: "'c" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "(\<forall>u. cinner x u = 0) = (x = zero)"
fin_digraphI_app_iso: fixes hom :: "('a, 'a \<times> nat \<times> 'a, 'c, 'd) digraph_isomorphism" assumes "H.digraph_isomorphism hom" shows "fin_digraph (app_iso hom (app_iso iso_f G))"
lfp_lowerbound: fixes L :: "'b set"   and Glb :: "'b set \<Rightarrow> 'b"   and a :: "'b"   and f :: "'b \<Rightarrow> 'b" assumes "Complete_Lattice L Glb"   and "a \<in> L"   and "f a \<le> a" shows "Complete_Lattice.lfp L Glb f \<le> a"
simps: fixes p :: "real poly"   and q :: "real poly" shows "sturm_aux p q = (if degree q = 0 then [p, q] else p # sturm_aux q (- (p mod q)))"
lex_ext_least_2: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> bool \<times> bool"   and m :: "nat"   and ys :: "'a list" shows "\<not> fst (lex_ext f m [] ys)"
smooth_on_add_fun: fixes k :: "enat"   and S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "k-smooth_on S f"   and "k-smooth_on S g"   and "open S" shows "k-smooth_on S (f + g)"
assms_4: fixes w :: "'a" assumes "v \<in> VV p**"   and "v \<rightarrow> w" shows "w \<in> S"
cball_diff_sphere: fixes a :: "'a"   and r :: "real" shows "cball a r - sphere a r = ball a r"
neg_closed: fixes g :: "'g" assumes "g \<in> G" shows "- g \<in> G"
Cauchy_snd: fixes X :: "nat \<Rightarrow> 'd \<times> 'e" assumes "Cauchy X" shows "Cauchy (\<lambda>n. snd (X n))"
RDirProd_list_def: fixes Rs :: "('c, 'd) ring_scheme list" shows "RDirProd_list Rs = foldr (\<lambda>R S. image_ring (\<lambda>(a, as). a # as) (RDirProd R S)) Rs nil_ring"
field_differentiable_compose_within: fixes f :: "'a \<Rightarrow> 'a"   and a :: "'a"   and S :: "'a set"   and g :: "'a \<Rightarrow> 'a" assumes "f field_differentiable at a within S"   and "g field_differentiable at (f a) within f ` S" shows "g \<circ> f field_differentiable at a within S"
indets_restrict_indets_subset: fixes p :: "('c option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "indets (restrict_indets p) \<subseteq> the ` (indets p - {None})"
simps_2: fixes l :: "('c, 'd) tree"   and k :: "'c"   and v :: "'d"   and r :: "('c, 'd) tree" shows "in_traverse_pairs (Node l k v r) = in_traverse_pairs l @ (k, v) # in_traverse_pairs r"
invar: fixes s :: "('a, unit) HashMap.hashmap" shows "my_invar s"
setops_SMP_cases: fixes t :: "('fun, 'var) Term.term"   and S :: "('fun, 'var) stateful_strand_step list" assumes "t \<in> SMP (pair ` setops\<^sub>s\<^sub>s\<^sub>t S)"   and "\<forall>p. Ana (pair p) = ([], [])" shows "(\<exists>\<delta>. wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>) \<and> t \<in> pair ` setops\<^sub>s\<^sub>s\<^sub>t S \<cdot>\<^sub>s\<^sub>e\<^sub>t \<delta>) \<or> t \<in> SMP (trms\<^sub>s\<^sub>s\<^sub>t S)"
1: shows "scut\<^sup>+\<^sup>+ p p"
dense_repr_set_fst: fixes p :: "'a list list"   and K :: "'a list set" assumes "set p \<subseteq> K" shows "fst ` set (UP.dense_repr p) \<subseteq> K - {\<zero>\<^bsub>K [X]\<^esub>}"
Loss_Delay_msg_a: fixes a :: "nat \<Rightarrow> aType list"   and x :: "nat \<Rightarrow> aType list"   and i1 :: "nat \<Rightarrow> ECall_Info list"   and d :: "nat"   and y :: "nat \<Rightarrow> aType list"   and i2 :: "nat \<Rightarrow> ECall_Info list"   and lose :: "nat \<Rightarrow> bool list"   and i :: "nat \<Rightarrow> ECall_Info list" assumes "msg (Suc 0) a"   and "Delay x i1 d y i2"   and "Loss lose a i2 x i" shows "msg (Suc 0) y"
defs': fixes v :: "'var \<times> 'node \<times> Def"   and g :: "'g"   and n :: "'node" shows "(v \<in> defs' g n) = (var g v \<in> defs g n \<and> defKind v = SimpleDef \<and> defNode v = n)"
mod_ring_rel32_def: fixes pp :: "uint32"   and x :: "uint32"   and y :: "'a mod_ring" assumes "p = int_of_uint32 pp"   and "p \<le> 65535" shows "mod_ring_rel32 x y = (\<exists>z. urel32 x z \<and> mod_ring_rel z y)"
distinct_sorted_mono: fixes l :: "'a list"   and i :: "nat"   and j :: "nat" assumes "sorted l"   and "distinct l"   and "i < j"   and "j < length l" shows "l ! i < l ! j"
yivp_3: shows "Y\<^sub>2 \<in> U \<rightarrow> S"
assms_3: shows "qf = (\<lambda>x y. a11 * x\<^sup>2 + 2 * a12 * x * y + a22 * y\<^sup>2 + b1 * x + b2 * y + c)"
gkernTr7: fixes F :: "('e, 'f) Group_scheme"   and G :: "('g, 'h) Group_scheme"   and f :: "'e \<Rightarrow> 'g" assumes "Group F"   and "Group G"   and "f \<in> gHom F G" shows "F \<guillemotright> gker\<^bsub>F,G\<^esub> f "
filter_M_not_more_elements: fixes h :: "'l"   and P :: "'n \<Rightarrow> ('l, 'm, bool) prog"   and xs :: "'n list"   and ys :: "'n list"   and x :: "'n" assumes "h \<turnstile> filter_M P xs \<rightarrow>\<^sub>r ys"   and "\<And>x. x \<in> set xs \<Longrightarrow> pure (P x) h"   and "x \<in> set ys" shows "x \<in> set xs"
OrdNotEqP_graph_aux_def: shows "OrdNotEqP_graph_aux \<equiv> lfp (\<lambda>p x1 x2. \<exists>OrdNotEqP_aux x y. x1 = (x, y) \<and> x2 = OrdP x AND OrdP y AND (x IN y OR y IN x))"
domain_export'': fixes x :: "'a"   and y :: "'a" shows "d (d x \<cdot> y) = d x \<cdot> d y"
fcard_finsert_disjoint: fixes x :: "'b"   and A :: "'b fset" assumes "x |\<notin>| A" shows "fcard (finsert x A) = Suc (fcard A)"
distinct_39: fixes x8 :: "'a" shows "Boolean \<noteq> \<langle>x8\<rangle>\<^sub>\<T>"
measure_pmf_of_set: fixes S :: "'b set"   and A :: "'b set" assumes "S \<noteq> {}"   and "finite S" shows "measure_pmf.prob (pmf_of_set S) A = real (card (S \<inter> A)) / real (card S)"
NEST_BEGIN: fixes i :: "'b" assumes "\<I> i" shows "NEST (BEGIN i)"
simps_2: fixes x :: "'b"   and xs :: "'b list"   and ys :: "'b list" shows "(x # xs) @ ys = x # xs @ ys"
B: shows "\<forall>x. z (Suc x) = (\<not> z x)"
abs_int_poly_def: fixes p :: "int poly" shows "abs_int_poly p \<equiv> if lead_coeff p < 0 then - p else p"
rep_eq: fixes x :: "'a Filters.filter"   and xa :: "'a Filters.filter" shows "(x < xa) = (Filters.filter.Rep_filter x \<subset> Filters.filter.Rep_filter xa)"
index_pref_pow_mod: fixes w :: "'b list"   and r :: "'b list"   and k :: "nat"   and i :: "nat" assumes "w \<le>p r \<^sup>@ k"   and "i < \<^bold>|w\<^bold>|" shows "w ! i = r ! (i mod \<^bold>|r\<^bold>|)"
le_imp_add_int: fixes i :: "nat"   and j :: "nat" assumes "i \<le> j" shows "\<exists>k. j = i + k"
eq_iff: fixes x :: "'a list"   and y :: "'a list" shows "(x = y) = (border x y \<and> border y x)"
eq2: shows "weight_spmf q \<le> weight_spmf p"
dvd_field_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and 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 b :: "'b" assumes "class.field divide inverse times one plus zero minus uminus" shows "dvd.dvd times a b = (a = zero \<longrightarrow> b = zero)"
items_le_paths_le: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and k :: "nat"   and Doc :: "character list"   and P :: "(symbol \<times> character list) list set" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel" shows "items_le k (LocalLexing.Gen \<NN> \<TT> \<RR> \<SS> Doc P) = LocalLexing.Gen \<NN> \<TT> \<RR> \<SS> Doc (paths_le k P)"
is_tangle_diagram_vert_block: fixes b :: "brick list" shows "is_tangle_diagram (b * basic (make_vert_block (nat (codomain_block b))))"
prems: shows "f' a = (f x \<circ> foldr f xs) (f' b)"
hyps_3: shows "count_weights True (remove_weights m1) = count_weights True (remove_weights m2)"
Pi_UN: fixes I :: "'d set"   and A :: "nat \<Rightarrow> 'd \<Rightarrow> 'e set" assumes "finite I"   and "\<And>i n m. \<lbrakk>i \<in> I; n \<le> m\<rbrakk> \<Longrightarrow> A n i \<subseteq> A m i" shows "(\<Union>n. Pi I (A n)) = (\<Pi> i\<in>I. \<Union>n. A n i)"
fcompare_double_Lt_trans: fixes x :: "double"   and y :: "double"   and z :: "double" assumes "fcompare_double x y = ccode.Lt"   and "fcompare_double y z = ccode.Lt" shows "fcompare_double x z = ccode.Lt"
nat_power_of_real_aux: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and nat_power :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and real_power :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and n :: "nat"   and x :: "real" assumes "class.nat_power_normed_field one times plus zero nat_power divide inverse minus uminus dist norm scaleR sgn uniformity open inner real_power" shows "nat_power n (scaleR x one) = scaleR (real n powr x) one"
MyhillNerode_def: fixes L :: "'b list set"   and R :: "('b list \<times> 'b list) set" shows "MyhillNerode L R \<equiv> equiv UNIV R \<and> right_invariant R \<and> finite (UNIV // R) \<and> (\<exists>A. L = R `` A)"
stepST_STCal_step: fixes P :: "('procS, 'procT) Proc"   and P' :: "('procS, 'procT) Proc" shows "P \<longmapsto>STCal Source Target P' = ((\<exists>S S'. S \<in>S P \<and> S' \<in>S P' \<and> S \<longmapsto>Source S') \<or> (\<exists>T T'. T \<in>T P \<and> T' \<in>T P' \<and> T \<longmapsto>Target T'))"
cong_mult_rcancel: fixes k :: "'b"   and m :: "'b"   and a :: "'b"   and b :: "'b" assumes "coprime k m" shows "[a * k = b * k] (mod m) = [a = b] (mod m)"
lookup_single_pp: fixes x :: "'c"   and e :: "'b"   and y :: "'c" shows "lookup_pp (single_pp x e) y = (e when x = y)"
has_contour_integral_integrable: fixes f :: "complex \<Rightarrow> complex"   and i :: "complex"   and g :: "real \<Rightarrow> complex" assumes "(f has_contour_integral i) g" shows "f contour_integrable_on g"
col_not_eq: shows "column i (fst (QR_decomposition A)) \<noteq> column ia (fst (QR_decomposition A))"
setmonotone_comp: fixes f :: "'b set \<Rightarrow> 'b set"   and g :: "'b set \<Rightarrow> 'b set" assumes "setmonotone f"   and "setmonotone g" shows "setmonotone (g \<circ> f)"
intro: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b" assumes "\<And>v. v \<notin> \<Union> X \<Longrightarrow> f v = v" shows "ChamberComplexAutomorphism_axioms X f"
orbit_swap: fixes s :: "'b"   and f :: "'b \<Rightarrow> 'b"   and t :: "'b" assumes "s \<in> orbit f s"   and "t \<in> orbit f s" shows "s \<in> orbit f t"
cexpr_typing_subst_aux: fixes x :: "nat"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and t :: "pdf_type"   and e' :: "cexpr"   and t' :: "pdf_type"   and e :: "cexpr" assumes "insert_var x \<Gamma> t \<turnstile>\<^sub>c e' : t'"   and "\<Gamma> \<turnstile>\<^sub>c e : t" shows "\<Gamma> \<turnstile>\<^sub>c cexpr_subst x e e' : t'"
infsetsum_mono_complex: fixes f :: "'c \<Rightarrow> complex"   and A :: "'c set"   and g :: "'c \<Rightarrow> complex" assumes "f abs_summable_on A"   and "g abs_summable_on A"   and "\<And>x. x \<in> A \<Longrightarrow> f x \<le> g x" shows "infsetsum f A \<le> infsetsum g A"
const_lt_component_SomeD1: fixes p :: "'t \<Rightarrow>\<^sub>0 'e"   and cmp :: "'k" assumes "const_lt_component p = Some cmp" shows "lp p = (0::'a)"
intros_1: fixes t :: "lam" shows " t \<longrightarrow>\<^sub>1\<^sup>* t"
HAInitState_def: fixes HA :: "('d, 'e, 'f) hierauto" shows "HAInitState HA \<equiv> InitState (HARoot HA)"
PO_l2_inv4': shows "reach l2 \<subseteq> l2_inv4'"
CastReds: fixes e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and C :: "char list" assumes "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>" shows "P \<turnstile> \<langle>Cast C e,s\<rangle> \<rightarrow>* \<langle>Cast C e',s'\<rangle>"
sem_def: fixes c :: "('c, 'd) conf" shows "sem c = (\<lambda>\<sigma>. \<sigma> \<in> states c)"
welltyped_App: fixes e1 :: "expr"   and T :: "Unary_PCF.type"   and U :: "Unary_PCF.type"   and e2 :: "expr" assumes "e1 ::: T \<rightarrow> U"   and "e2 ::: T" shows "e1 \<cdot> e2 ::: U"
elims_3: fixes x :: "'a extC"   and xa :: "'a extC" assumes "\<not> x \<le> xa" obtains uv_ :: "'a" where "xa__ = extC.Val uv_" and "xa = extC.Bot"   | x :: "'a" and y :: "'a" where "xa__ = extC.Val x" and "xa = extC.Val y" and "\<not> x \<le> y"   | "xa__ = extC.Top" and "xa = extC.Bot"   | ux_ :: "'a" where "xa__ = extC.Top" and "xa = extC.Val ux_"
sel_16: fixes x61 :: "'b mrexp3"   and x62 :: "'b set" shows "nul (Star3 x61 x62) = True"