hcju/isabelleps huggingface dataset:
b: shows "(c::'a::refinement_lattice) ; \<Sqinter> ((d::'b::type \<Rightarrow> 'a::refinement_lattice) ` (X::'b::type set)) = \<Sqinter> ((;) c ` d ` X)"
se_step: fixes prb :: "('d, 'e, 'f) pre_RedBlack"   and re :: "('d \<times> nat) edge"   and prb' :: "('d, 'e, 'f) pre_RedBlack"   and p' :: "('e, 'f) conf" assumes "RedBlack prb"   and "ui_edge re \<in> edges (black prb) \<and> (src re \<in> red_vertices prb \<and> tgt re \<notin> red_vertices prb \<and> red prb' = add_edge (red prb) re) \<and> src re \<notin> subsumees (subs prb) \<and> se (confs prb (src re)) (labelling (black prb) (ui_edge re)) p' \<and> prb' = \<lparr>red = red prb', black = black prb, subs = subs prb, init_conf = init_conf prb, confs = (confs prb)(tgt re := p'), marked = (marked prb)(tgt re := marked prb (src re)), strengthenings = strengthenings prb\<rparr>" shows "RedBlack prb'"
pre': shows "\<forall>s. ((\<lambda>a. I a \<and> vars b \<subseteq> dom (fst a) \<and> pbval b (fst a)) \<and>* F) s \<longrightarrow> pre C' ((I \<and>* $ (Suc 0)) \<and>* F) s"
le_rel: fixes D :: "('c, 'd) Order_scheme"   and a :: "'c"   and b :: "'c" assumes "Order D"   and "a \<in> carrier D"   and "b \<in> carrier D" shows "a \<preceq>\<^bsub>D\<^esub> b = ((a, b) \<in> rel D)"
sum_index_enum_eq: fixes f :: "nat \<Rightarrow> 'a" shows "(\<Sum>k\<in>UNIV. f (index enum_class.enum k)) = sum f {..<CARD('b)}"
inline_parametric': shows "rel_fun (rel_fun (=) (rel_fun (BNF_Def.Grp UNIV g) (rel_gpv'' (rel_prod (BNF_Def.Grp UNIV h)\<inverse>\<inverse> (=)) (=) (=)))) (rel_fun (rel_gpv'' (BNF_Def.Grp UNIV f) (BNF_Def.Grp UNIV g) (BNF_Def.Grp UNIV h)\<inverse>\<inverse>) (rel_fun (=) (rel_gpv'' (rel_prod (BNF_Def.Grp UNIV f) (=)) (=) (=)))) inline inline"
contractible_space_retraction_map_image: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c" assumes "retraction_map X Y f"   and "contractible_space X" shows "contractible_space Y"
hom_add: fixes x :: "'c poly poly"   and y :: "'c poly poly"   and xa :: "'c"   and ya :: "'c" shows "poly2 (x + y) xa ya = poly2 x xa ya + poly2 y xa ya"
map_iterator_linord_I: fixes l0 :: "('a \<times> 'd) list"   and m :: "'a \<Rightarrow> 'd option"   and iti :: "('e \<Rightarrow> bool) \<Rightarrow> ('a \<times> 'd \<Rightarrow> 'e \<Rightarrow> 'e) \<Rightarrow> 'e \<Rightarrow> 'e" assumes "distinct (map fst l0)"   and "m = map_of l0"   and "sorted (map fst l0)"   and "iti = foldli l0" shows "map_iterator_linord iti m"
mutual_information_def: fixes b :: "real"   and S :: "'c measure"   and T :: "'d measure"   and X :: "'b \<times> 'b \<Rightarrow> 'c"   and Y :: "'b \<times> 'b \<Rightarrow> 'd" shows "P.mutual_information b S T X Y = KL_divergence b (distr P S X \<Otimes>\<^sub>M distr P T Y) (distr P (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)))"
3: shows "- (- - x \<sqinter> - - y) \<sqinter> - x = - x"
inf_chain_def: fixes p :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'a" shows "inf_chain p f = (\<forall>i. p (f i) (f (Suc i)))"
intro: fixes ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "ext ext"   and "ext_total_axioms ext" shows "ext_total ext"
graph: shows "\<alpha> \<in> S \<rightarrow>\<^sub>E S"
l_neg: fixes x :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P" shows "\<ominus>\<^bsub>P\<^esub> x \<oplus>\<^bsub>P\<^esub> x = \<zero>\<^bsub>P\<^esub>"
421: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- - (x \<sqinter> y) \<sqinter> (- x \<sqinter> z) = bot"
append_Nil: fixes ys :: "'e list" shows "[] @ ys = ys"
less_cSupE: fixes y :: "'a"   and X :: "'a set" assumes "y < Sup X"   and "X \<noteq> {}" obtains x :: "'a" where "x \<in> X" and "y < x"
inf_Inf: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> y = \<Sqinter> insert x (\<eta> y)"
R1: fixes p :: "'b fm"   and q :: "'b fm" assumes "\<turnstile> p"   and "\<turnstile> (p \<^bold>\<longrightarrow> q)" shows "\<turnstile> q"
scale_minus_left: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'b"   and x :: "'c" assumes "module scale" shows "scale (- a) x = - scale a x"
steps_z_alpha_alt: fixes X :: "'d set"   and k :: "'d \<Rightarrow> nat"   and A :: "('e \<times> ('d, real) cconstraint \<times> 'f \<times> 'd list \<times> 'e) set \<times> ('e \<Rightarrow> ('d, real) cconstraint)"   and l :: "'e"   and Z :: "('d \<Rightarrow> real) set"   and l' :: "'e"   and Z' :: "('d \<Rightarrow> real) set"   and l'' :: "'e"   and Z'' :: "('d \<Rightarrow> real) set" assumes "AlphaClosure X"   and "AlphaClosure.step_z_alpha X k A l Z l' Z'"   and "AlphaClosure.steps_z_alpha X k A l' Z' l'' Z''" shows "AlphaClosure.steps_z_alpha X k A l Z l'' Z''"
assms: shows "nres_of S \<le> SPEC \<Phi>"
C5: fixes x :: "'b \<Longrightarrow> 'a" shows "(&x =\<^sub>u &x) = true"
not_eq_last_diff_paths_reachability_s: fixes s :: "('a, bool) fmap"   and PROB :: "(('a, bool) fmap \<times> ('a, bool) fmap) set"   and as :: "(('a, bool) fmap \<times> ('a, bool) fmap) list" assumes "s \<in> valid_states PROB"   and "as \<in> valid_plans PROB"   and "\<not> FactoredSystem.inj last (state_set (state_list s as)) (reachable_s PROB s)" shows "\<exists>slist_1 slist_2. slist_1 \<in> state_set (state_list s as) \<and> slist_2 \<in> state_set (state_list s as) \<and> last slist_1 = last slist_2 \<and> length slist_1 \<noteq> length slist_2"
map_le_antisym: fixes f :: "'c \<Rightarrow> 'd option"   and g :: "'c \<Rightarrow> 'd option" assumes "f \<subseteq>\<^sub>m g"   and "g \<subseteq>\<^sub>m f" shows "f = g"
going_to_def: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'b filter" shows "f going_to F = filtercomap f F"
simps: fixes x :: "'e"   and g :: "'f \<Rightarrow> ('e, 'f) step"   and b :: "bool"   and s :: "'f" shows "remove1_raw x g (b, s) = (case g s of Done \<Rightarrow> Done | Skip s' \<Rightarrow> Skip (b, s') | Yield y s' \<Rightarrow> if b \<and> x = y then Skip (False, s') else Yield y (b, s'))"
gcp_lb1: fixes xs :: "'b list"   and ys :: "'b list" shows "prefix (gcp xs ys) xs"
fg: fixes x :: "'a" assumes "x \<in> S \<and> \<not> P x \<or> x \<in> T \<and> P x" shows "f x = g x"
ker_ideal: fixes A :: "('c, 'd) Ring_scheme"   and R :: "('e, 'f) Ring_scheme"   and f :: "'c \<Rightarrow> 'e" assumes "Ring A"   and "Ring R"   and "f \<in> rHom A R" shows "ideal A (ker\<^bsub>A,R\<^esub> f)"
G_add_edge_def: shows "G_add_edge \<equiv> wgraph_by_map.add_edge RBT_Map.update lookup"
lookup_cases: fixes y :: "'a \<Rightarrow> 'b" assumes "y \<in> {f. finite {x. f x \<noteq> (0::'b)}}" obtains x :: "'a \<Rightarrow>\<^sub>0 'b" where "y = poly_mapping.lookup x"
evals_ts: fixes G :: "prog"   and s :: "state"   and es :: "expr list"   and vs :: "val list"   and s' :: "state"   and L :: "lenv"   and C :: "qtname"   and Ts :: "ty list"   and A :: "assigned" assumes "G\<turnstile>s \<midarrow>es\<doteq>\<succ>vs\<rightarrow> s'"   and "wf_prog G"   and "s\<Colon>\<preceq>(G, L)"   and "\<lparr>prg = G, cls = C, lcl = L\<rparr>,empty_dt\<Turnstile>es\<Colon>\<doteq>Ts"   and "\<lparr>prg = G, cls = C, lcl = L\<rparr>\<turnstile> dom (locals (snd s)) \<guillemotright>In3 es\<guillemotright> A" shows "s'\<Colon>\<preceq>(G, L) \<and> (normal s' \<longrightarrow> list_all2 (conf G (snd s')) vs Ts) \<and> error_free s = error_free s'"
hypnat_diff_add_inverse: fixes n :: "nat star"   and m :: "nat star" shows "n + m - n = m"
ASSUME_eq: shows "ASSUME \<equiv> iASSUME return"
S_selects_subseteq: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset"   and C :: "'b literal multiset" assumes "selection S" shows "S C \<subseteq># C"
preserves_inv: fixes f :: "'c" assumes "C.iso f" shows "C'.T (C.cod \<iota>\<^sub>C, C.inv f) = C.inv (C'.T (C.cod \<iota>\<^sub>C, f))"
MatchOr: fixes \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and m1 :: "'c match_expr"   and m2 :: "'c match_expr"   and a :: "action"   and p :: "'d" shows "matches \<gamma> (MatchOr m1 m2) a p = (matches \<gamma> m1 a p \<or> matches \<gamma> m2 a p)"
poly_squarefree_decomp: fixes p :: "'a poly"   and q :: "'a poly"   and d :: "'a poly"   and e :: "'a poly"   and r :: "'a poly"   and s :: "'a poly" assumes "pderiv p \<noteq> 0"   and "p = q * d"   and "pderiv p = e * d"   and "d = r * p + s * pderiv p" shows "rsquarefree q \<and> (\<forall>a. (poly q a = (0::'a)) = (poly p a = (0::'a)))"
finite_maxsimp: fixes x :: "'a set" assumes "domain.chamber x" shows "finite x"
minus_pred_def: fixes P :: "'a pred"   and Q :: "'a pred" shows "P - Q = Pred (eval P - eval Q)"
Quotient_ocircline: shows "Quotient ocircline_eq_clmat abs_ocircline rep_ocircline cr_ocircline"
T2': fixes y :: "'c"   and x :: "'c" assumes "R2 y x" shows "T2 x (Abs2 y)"
issuedp_def: fixes evs :: "event list"   and k :: "key" shows "issuedp evs k = (k \<in> issued evs)"
hcomp_in_vhom: fixes \<mu> :: "'d"   and f :: "'d"   and g :: "'d"   and \<nu> :: "'d"   and h :: "'d"   and k :: "'d" assumes "\<guillemotleft>\<mu> : f \<Rightarrow>\<^sub>D g\<guillemotright>"   and "\<guillemotleft>\<nu> : h \<Rightarrow>\<^sub>D k\<guillemotright>"   and "src\<^sub>D h = trg\<^sub>D f" shows "\<guillemotleft>\<nu> \<star>\<^sub>D \<mu> : h \<star>\<^sub>D f \<Rightarrow>\<^sub>D k \<star>\<^sub>D g\<guillemotright>"
nat_mult_1: fixes n :: "nat" shows "1 *n n = n"
orthogonal: fixes i :: "nat"   and j :: "nat" assumes "i < length (map of_int_hom.vec_hom fs')"   and "j < length (map of_int_hom.vec_hom fs')"   and "i \<noteq> j" shows "gs2.gso i \<bullet> gs2.gso j = 0"
iso_fiter: fixes f :: "'b \<Rightarrow> 'b" assumes "mono f" shows "mono (fiter_id f)"
map_comp: fixes f :: "'i \<Rightarrow> 'h"   and g :: "'j \<Rightarrow> 'i"   and p :: "'j spmf" shows "map_spmf f (map_spmf g p) = map_spmf (f \<circ> g) p"
mult_2: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c" assumes "class.semiring_numeral one times plus" shows "times (numeral.numeral one plus (num.Bit0 num.One)) z = plus z z"
times_def: fixes x :: "'c"   and y :: "'c" shows "x * y = x \<sqinter> y"
mix_of_not_preferred_is_not_preferred: fixes w :: "'a pmf"   and p :: "'a pmf"   and q :: "'a pmf"   and \<alpha> :: "real" assumes "w \<succeq>[\<R>] p"   and "w \<succeq>[\<R>] q"   and "\<alpha> \<in> {0..1}" shows "w \<succeq>[\<R>] mix_pmf \<alpha> p q"
equivclp_refl: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d" shows "equivclp r x x"
continuous_map_of_fst: fixes X :: "'d topology"   and Y :: "'e topology"   and Z :: "'f topology"   and f :: "'d \<Rightarrow> 'f" shows "continuous_map (prod_topology X Y) Z (f \<circ> fst) = (topspace Y = {} \<or> continuous_map X Z f)"
M: shows "\<forall>m\<in>M. \<forall>f T. Fun f T \<sqsubseteq> m \<longrightarrow> P (Fun f T)"
ennreal_sum: fixes A :: "'b set"   and f :: "'b \<Rightarrow> real" assumes "finite A"   and "\<And>a. 0 \<le> f a" shows "(\<Sum>a\<in>A. ennreal (f a)) = ennreal (sum f A)"
c_1: fixes x :: "real" assumes "x\<^sub>0 \<le> x"   and "x \<le> x\<^sub>1" shows "c \<le> f_approx x"
openin_euclidean_subtopology_iff: fixes U :: "'a set"   and S :: "'a set" shows "openin (top_of_set U) S = (S \<subseteq> U \<and> (\<forall>x\<in>S. \<exists>e>0. \<forall>x'\<in>U. dist x' x < e \<longrightarrow> x' \<in> S))"
disjI2: fixes \<gamma>' :: "(nat \<Rightarrow> cnf) \<Rightarrow> nat \<Rightarrow> bool"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and \<gamma> :: "(nat \<Rightarrow> cnf) \<Rightarrow> nat \<Rightarrow> bool" assumes "\<gamma>' t n" shows "(\<gamma> \<or>\<^sup>c \<gamma>') t n"
fail_converter_sel: fixes a :: "'b" shows "run_converter fail_converter a = Fail"
game_1_def: fixes \<A> :: "(bool, bool list, bool list) gpv" shows "PRF.game_1 \<A> = exec_gpv PRF.random_oracle \<A> Map.empty \<bind> (\<lambda>(b, uu_). return_spmf b)"
that: fixes a :: "V"   and b :: "V" assumes "x = \<langle>a, b\<rangle>"   and "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r" shows "thesis"
cases: fixes a :: "'c"   and g :: "'c \<Rightarrow> ('d, 'c) step" assumes "a \<in> terminates_on g" obtains s :: "'c" where "a = sa__" and "g sa__ = Done"   | s :: "'c" and s' :: "'c" where "a = sb__" and "g sb__ = Skip s'__" and "s'__ \<in> terminates_on g"   | s :: "'c" and aa :: "'d" and s' :: "'c" where "a = s" and "g s = Yield aa s'" and "s' \<in> terminates_on g"
list_slice2_le: fixes xs :: "'b list"   and k :: "nat" assumes "length xs \<le> k" shows "list_slice2 xs k = (if length xs = 0 then [] else [xs])"
smult_map_poly: fixes a :: "'a" shows "smult a = map_poly ((*) a)"
simps_4: fixes r :: "'a set"   and b :: "'a set"   and c :: "'a ann_com" shows "atomics (AnnCond2 r b c) = atomics c"
InfinitesimalD2: fixes x :: "'a star"   and r :: "real" assumes "x \<in> Infinitesimal"   and "0 < r" shows "hnorm x < hypreal_of_real r"
assms_2: shows "continuous_on (path_image \<gamma>) (\<lambda>x. F x \<bullet> b)"
sec_id_pubk_less_msg: fixes m :: "nat"   and m' :: "nat"   and n :: "nat" assumes "{m, m'} \<subseteq> bad_sec_prik"   and "(n, m) \<notin> bad_id"   and "(n, m') \<in> bad_id" shows "{Sec_PriKey m, ID n (Sec_PubKey m), Hash (ID n (Sec_PubKey m)), \<langle>n, PubKey {Tok_PriK n, Rev_PriK}\<rangle>} \<subseteq> spied (sec_id_pubk_less n) \<and> {\<langle>n, ID n (Sec_PubKey m)\<rangle>, \<langle>n, Hash (ID n (Sec_PubKey m))\<rangle>} \<inter> spied (sec_id_pubk_less n) = {}"
sub_ssI: fixes \<V> :: "'b set"   and \<B> :: "'b set multiset"   and \<U> :: "'b set"   and \<A> :: "'b set multiset" assumes "incidence_system \<V> \<B>"   and "\<U> \<subseteq> \<V>"   and "\<A> \<subseteq># \<B>" shows "sub_set_system \<U> \<A> \<V> \<B>"
Least1I: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool" assumes "class.order less_eq less"   and "\<exists>!x. P x \<and> (\<forall>y. P y \<longrightarrow> less_eq x y)" shows "P (ord.Least less_eq P)"
sgn_minus: fixes a :: "'b" shows "sgn (- a) = - sgn a"
append_eq_append_conv: fixes xs :: "'k list"   and ys :: "'k list"   and us :: "'k list"   and vs :: "'k list" assumes "length xs = length ys \<or> length us = length vs" shows "(xs @ us = ys @ vs) = (xs = ys \<and> us = vs)"
induct: fixes P :: "WS1S_Prelim.order \<Rightarrow> 'a \<Rightarrow> ('a, 'a) atomic \<Rightarrow> bool"   and a0 :: "WS1S_Prelim.order"   and a1 :: "'a"   and a2 :: "('a, 'a) atomic" assumes "\<And>i m. P FO i (Fo m)"   and "\<And>i uu_ m uv_. P FO i (Eq_Const uu_ m uv_)"   and "\<And>i uw_ m1 m2. P FO i (Less uw_ m1 m2)"   and "\<And>i ux_ m1 m2 uy_. P FO i (Plus_FO ux_ m1 m2 uy_)"   and "\<And>i uz_ m1 m2. P FO i (Eq_FO uz_ m1 m2)"   and "\<And>i M1 M2. P SO i (Eq_SO M1 M2)"   and "\<And>i va_ vb_ M1 M2. P SO i (Suc_SO va_ vb_ M1 M2)"   and "\<And>i M. P SO i (Empty M)"   and "\<And>i M. P SO i (Singleton M)"   and "\<And>i M1 M2. P SO i (Subset M1 M2)"   and "\<And>i vc_ m vd_. P FO i (In vc_ m vd_)"   and "\<And>i ve_ vf_ M. P SO i (In ve_ vf_ M)"   and "\<And>i vg_ m vh_. P FO i (Eq_Max vg_ m vh_)"   and "\<And>i vi_ vj_ M. P SO i (Eq_Max vi_ vj_ M)"   and "\<And>i vk_ m vl_. P FO i (Eq_Min vk_ m vl_)"   and "\<And>i vm_ vn_ M. P SO i (Eq_Min vm_ vn_ M)"   and "\<And>i M1 M2 M3. P SO i (Eq_Union M1 M2 M3)"   and "\<And>i M1 M2 M3. P SO i (Eq_Inter M1 M2 M3)"   and "\<And>i M1 M2 M3. P SO i (Eq_Diff M1 M2 M3)"   and "\<And>i M1 M2. P SO i (Disjoint M1 M2)"   and "\<And>i vo_ M vp_. P SO i (Eq_Presb vo_ M vp_)"   and "\<And>vr_ v. P SO vr_ (Fo v)"   and "\<And>vr_ v va vb. P SO vr_ (Eq_Const v va vb)"   and "\<And>vr_ v va vb. P SO vr_ (Less v va vb)"   and "\<And>vr_ v va vb vc. P SO vr_ (Plus_FO v va vb vc)"   and "\<And>vr_ v va vb. P SO vr_ (Eq_FO v va vb)"   and "\<And>vr_ v va vb. P SO vr_ (Eq_Const v va vb)"   and "\<And>vr_ v va vb. P SO vr_ (Less v va vb)"   and "\<And>vr_ v va vb vc. P SO vr_ (Plus_FO v va vb vc)"   and "\<And>vr_ v va vb. P SO vr_ (Eq_FO v va vb)"   and "\<And>vr_ v va. P FO vr_ (Eq_SO v va)"   and "\<And>vr_ v va vb vc. P FO vr_ (Suc_SO v va vb vc)"   and "\<And>vr_ v. P FO vr_ (Empty v)"   and "\<And>vr_ v. P FO vr_ (Singleton v)"   and "\<And>vr_ v va. P FO vr_ (Subset v va)"   and "\<And>vr_ v va vb. P FO vr_ (Eq_Union v va vb)"   and "\<And>vr_ v va vb. P FO vr_ (Eq_Inter v va vb)"   and "\<And>vr_ v va vb. P FO vr_ (Eq_Diff v va vb)"   and "\<And>vr_ v va. P FO vr_ (Disjoint v va)"   and "\<And>vr_ v va vb. P FO vr_ (Eq_Presb v va vb)" shows "P a0 a1 a2"
diaL: fixes A :: "(C, BD) form" shows "g3mod1p [ \<LM> A  \<RM> \<Rightarrow>* \<Empt>] ( \<LM> \<diamond> A  \<RM> \<Rightarrow>* \<Empt>)"
vunion_vempty_left: fixes a :: "V" shows "0 \<union>\<^sub>\<circ> a = a"
induced_morphism_with_id_is_local: fixes x :: "'a"   and V :: "'a set" assumes "x \<in> S"   and "x \<in> V"   and "is_open V" shows "ind_mor_btw_stalks.is_local S is_open \<FF> \<rho> add_str mult_str zero_str one_str is_open \<FF> \<rho> add_str mult_str zero_str one_str (identity S) x V \<phi>\<^bsub>S is_open \<FF> \<rho> is_open \<FF> \<rho> (identity S) (\<lambda>U. identity (\<FF> U)) x\<^esub>"
prems_6: shows "(ae_, a'e_) \<in> \<langle>R\<rangle>ltlr_rel \<rightarrow> \<langle>R\<rangle>ltlr_rel \<rightarrow> Rv \<rightarrow> Rv \<rightarrow> Rv"
span_explicit: fixes b :: "('a \<Rightarrow>\<^sub>0 'b) set" shows "phull b = {\<Sum>a\<in>t. r a \<cdot> a |t r. finite t \<and> t \<subseteq> b}"
simps_1: fixes w :: "nat"   and a :: "'a" shows "cachedWeight (Leaf w a) = w"
option_of_def: shows "option_of \<equiv> (\<lambda>(option, g). (Abs_game option, Abs_game g)) ` is_option_of"
ideal_prod_in_carrier: fixes R :: "('c, 'd) ring_scheme"   and I :: "'c set"   and J :: "'c set" assumes "ring R"   and "ideal I R"   and "ideal J R" shows "I \<cdot>\<^bsub>R\<^esub> J \<subseteq> carrier R"
Lp_subalgebra_4: fixes M :: "'b measure"   and F :: "'b measure"   and f :: "'b \<Rightarrow> real"   and p :: "ennreal" assumes "subalgebra M F"   and "f \<in> space\<^sub>N (\<LL> p (restr_to_subalg M F))" shows "Norm (\<LL> p M) f = Norm (\<LL> p (restr_to_subalg M F)) f"
is_lub_is_type: fixes wf_md :: "'b prog \<Rightarrow> String.literal \<Rightarrow> 'b mdecl \<Rightarrow> bool"   and P :: "'b prog"   and T :: "ty"   and U :: "ty"   and V :: "ty" assumes "wf_prog wf_md P"   and "is_type P T"   and "is_type P U"   and "P \<turnstile> lub(T, U) = V" shows "is_type P V"
Solution_imp_cond_D: fixes a :: "nat list"   and b :: "nat list"   and x :: "nat list"   and y :: "nat list" assumes "hlde a b"   and "(x, y) \<in> Linear_Diophantine_Equations.Solutions a b" shows "hlde.cond_D a b x y"
allocationPowerset: fixes a :: "('a \<times> 'b set) set"   and G :: "'b set"   and N :: "'a set" assumes "a \<in> allAllocations N G" shows "a \<subseteq> N \<times> (Pow G - {{}})"
simps_2: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and x1 :: "'b"   and x2 :: "'c" shows "(case (x1, x2) of (x, xa) \<Rightarrow> f x xa) = f x1 x2"
unat_add_lem': fixes x :: "'b word"   and y :: "'b word" assumes "unat x + unat y < 2 ^ LENGTH('b)" shows "unat (x + y) = unat x + unat y"
sets_count_space: fixes \<Omega> :: "'b set" shows "sets (count_space \<Omega>) = Pow \<Omega>"
fp_cop_F_worst: fixes x :: "'a"   and ds :: "'b set"   and y :: "'a" assumes "x \<in> cop ds"   and "y \<in> fp_cop_F ds"   and "Xd y = Xd x" shows "(x, y) \<in> Pd (Xd x)"
rquot_dfs_invariant: fixes x :: "nat" assumes "dfa_is_node A x" shows "rquot_invariant A (rquot_dfs A n x)"
axiom_not_def: fixes p :: "fol fm" shows "axiom_not p \<equiv> Thm (Iff (fm.Not p) (Imp p Falsity))"
isOK_forallM_index: fixes P :: "'a \<Rightarrow> nat \<Rightarrow> 'b + unit"   and xs :: "'a list" shows "isOK (forallM_index P xs) = (\<forall>i<length xs. isOK (P (xs ! i) i))"
pratt_triple_1: fixes p :: "nat"   and x :: "nat"   and a :: "nat" assumes "1 < p"   and "x = 1" shows "pratt_triple p a x"
dirichlet_prod'_assoc: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a"   and h :: "real \<Rightarrow> 'a"   and x :: "real" shows "dirichlet_prod' f (dirichlet_prod' g h) x = dirichlet_prod' (dirichlet_prod f g) h x"
CallRec: fixes P :: "'d set"   and p :: "'e"   and Q :: "'d set"   and A :: "'d set"   and Specs :: "('d set \<times> 'e \<times> 'd set \<times> 'd set) set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "('d set \<times> 'e \<times> 'd set \<times> 'd set) set"   and F :: "'f set" assumes "(P, p, Q, A) \<in> Specs"   and "\<forall>(P, p, Q, A)\<in>Specs. p \<in> dom \<Gamma> \<and> \<Gamma>,\<Theta> \<union> Specs\<turnstile>\<^bsub>/F \<^esub>P the (\<Gamma> p) Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P Call p Q,A"
A'_def: shows "A' = row_add A (mod_type_class.from_nat (Suc n)) i (- A $ mod_type_class.from_nat (Suc n) $ j)"
C: fixes s :: "complex" assumes "s \<in> U" shows "cmod (pochhammer s (Suc (2 * n))) \<le> C"
fst: fixes A :: "'b ptrm"   and B :: "'b ptrm" assumes "A \<approx> B" shows "PFst A \<approx> PFst B"
functional_relation: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "right_unique R"   and "left_total R" shows "\<forall>x. \<exists>!y. R x y"
gt_compat_arg: fixes s' :: "('s, 'v) tm"   and s :: "('s, 'v) tm"   and t :: "('s, 'v) tm" assumes "s' >\<^sub>t s" shows "App s' t >\<^sub>t App s t"
g_def: fixes x :: "'b" shows "g x = (SOME y. y \<in> s \<and> f y = x)"
ldistinct_lfilterI: fixes lst :: "'b llist"   and P :: "'b \<Rightarrow> bool" assumes "\<And>i j. \<lbrakk>enat i < llength lst; enat j < llength lst; lst $ i = lst $ j; P (lst $ i)\<rbrakk> \<Longrightarrow> i = j" shows "ldistinct (lfilter P lst)"
get_element_by_id_def: shows "Shadow_DOM.get_element_by_id \<equiv> l_get_element_by\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_element_by_id Shadow_DOM.first_in_tree_order get_attribute"
maxpositions_def: fixes l :: "'c list" shows "maxpositions l = filterpositions2 ((\<le>) (Max (set l))) l"
T_def: shows "T = (\<lambda>a. if a \<in> FG then a \<cdot> v else (0::'c))"
rk_ax_3_pts: fixes A :: "Points"   and B :: "Points" shows "\<exists>C. rk {A, B, C} = 2 \<and> rk {B, C} = 2 \<and> rk {A, C} = 2"
aform_of_ivl_def: fixes l :: "'a"   and u :: "'a" shows "aform_of_ivl l u = ((l + u) /\<^sub>R 2, pdevs_of_ivl l u)"
ideal_zero_forall: fixes R :: "('c, 'd) Ring_scheme" assumes "Ring R" shows "\<forall>I. ideal R I \<longrightarrow> \<zero>\<^bsub>R\<^esub> \<in> I"
r\<^sub>0s\<^sub>1_is_cospan: shows "cospan_of_maps_in_bicategory_of_spans (\<cdot>) (\<star>) \<a> \<i> src trg (tab\<^sub>1 h) (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)"
distance_lengths_finite: fixes v :: "'a"   and w :: "'a" shows "finite {length xs |xs. v \<leadsto>xs\<leadsto> w}"
dfs_\<alpha>_def: fixes S :: "'b set \<times> 'b list" shows "dfs_\<alpha> S \<equiv> let (\<Sigma>, W) = S in (\<Sigma>, set W)"
frombd_F13_tobd_F13: fixes b :: "'d"   and x :: "('e, 'f, 'd) F1"   and r2 :: "(bd_type_F2 \<times> bd_type_F2) set" assumes "b \<in> F1set3 x" shows "fromCard (F1set3 x) (dir_image (bd_F1 +c r2) Abs_bd_type_F) (toCard (F1set3 x) (dir_image (bd_F1 +c r2) Abs_bd_type_F) b) = b"
set_iterator_linord_I: fixes l0 :: "'a list"   and S :: "'a set"   and iti :: "('c \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'c" assumes "distinct l0"   and "S = set l0"   and "sorted l0"   and "iti = foldli l0" shows "set_iterator_linord iti S"
unit_simps_5: fixes a :: "'c" assumes "C.obj a" shows "D.cod (G.unit a) = G a"
fl: shows "(f \<longlongrightarrow> l) (at t within {..t})"
cauchy_const: fixes x :: "rat" shows "Real.cauchy (\<lambda>n. x)"
sub_induction: fixes u :: "lam"   and \<tau> :: "ty"   and t :: "lam"   and \<sigma> :: "ty" assumes "SN u"   and "u \<in> RED \<tau>"   and "NEUT t"   and "CR2 \<tau>"   and "CR3 \<sigma>"   and "CR3_RED t (\<tau> \<rightarrow> \<sigma>)" shows "App t u \<in> RED \<sigma>"
split_sel: fixes P :: "'g \<Rightarrow> bool"   and f1 :: "'g"   and f2 :: "'h \<Rightarrow> 'h list \<Rightarrow> 'g"   and list :: "'h list" shows "P (case list of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = ((list = [] \<longrightarrow> P f1) \<and> (list = hd list # tl list \<longrightarrow> P (f2 (hd list) (tl list))))"