hcju/isabelleps huggingface dataset:
BinOpThrow1: fixes 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 e\<^sub>1 :: "char list exp"   and s\<^sub>0 :: "(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\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and bop :: "bop"   and e\<^sub>2 :: "char list exp" assumes "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e,s\<^sub>1\<rangle>" shows "P \<turnstile> \<langle>e\<^sub>1 \<guillemotleft>bop\<guillemotright> e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e,s\<^sub>1\<rangle>"
res: shows "local.select_correct_factor_main a todo old l r n = ((q, ri_fin), l_fin, r_fin)"
app_red_cong1: fixes e1 :: "exp"   and e1' :: "exp"   and e2 :: "exp" assumes "e1 \<longrightarrow>* e1'" shows "EApp e1 e2 \<longrightarrow>* EApp e1' e2"
naturality_hom_induced: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and T :: "'b set"   and q :: "int" assumes "continuous_map X Y f"   and "f ` S \<subseteq> T" shows "hom_boundary q Y T \<circ> hom_induced q X S Y T f = hom_induced (q - 1) (subtopology X S) {} (subtopology Y T) {} f \<circ> hom_boundary q X S"
pos_zmult_eq_1_iff_lemma: fixes m :: "int"   and n :: "int" assumes "m * n = 1" shows "m = 1 \<or> m = - 1"
dlet3: fixes ck :: "bool"   and env :: "v sem_env"   and s1 :: "'b state"   and e :: "exp0"   and s2 :: "'b state"   and v1 :: "v"   and p :: "pat"   and mn :: "char list list"   and locs :: "locn \<times> locn" assumes "evaluate ck env s1 e (s2, Rval v1) \<and> allDistinct (pat_bindings p []) \<and> pmatch (c env) (refs s2) p v1 [] = Match_type_error" shows "evaluate_dec ck mn env s1 (Dlet locs p e) (s2, Rerr (Rabort Rtype_error))"
grid_base_out: fixes d :: "nat"   and dm :: "nat"   and ds :: "nat set"   and b :: "(nat \<times> int) list"   and p :: "(nat \<times> int) list" assumes "d < dm"   and "d \<notin> ds"   and "b \<in> sparsegrid' dm"   and "p \<in> grid (base ds b) ds" shows "p ! d = b ! d"
beta_induct: fixes t :: "sterm"   and t' :: "sterm"   and P1 :: "sterm \<Rightarrow> sterm \<Rightarrow> bool"   and P2 :: "sterm \<Rightarrow> sterm \<Rightarrow> bool" assumes "t -> t'"   and "\<And>t t' u l. \<lbrakk>t -> t'; P1 t t'; lc u\<rbrakk> \<Longrightarrow> P1 (Call t l u) (Call t' l u)"   and "\<And>t t' u l. \<lbrakk>t -> t'; P1 t t'; lc u\<rbrakk> \<Longrightarrow> P1 (Call u l t) (Call u l t')"   and "\<And>t t' u l. \<lbrakk>t -> t'; P1 t t'; body u\<rbrakk> \<Longrightarrow> P1 (Upd t l u) (Upd t' l u)"   and "\<And>t t' u l. \<lbrakk>P2 t t'; lc u\<rbrakk> \<Longrightarrow> P1 (Upd u l t) (Upd u l t')"   and "\<And>l f T t. \<lbrakk>l \<in> dom f; lc (Obj f T); body t\<rbrakk> \<Longrightarrow> P1 (Upd (Obj f T) l t) (Obj (f(l \<mapsto> t)) T)"   and "\<And>l f t t' T. \<lbrakk>l \<in> dom f; P2 t t'; \<forall>l\<in>dom f. body (the (f l))\<rbrakk> \<Longrightarrow> P1 (Obj (f(l \<mapsto> t)) T) (Obj (f(l \<mapsto> t')) T)"   and "\<And>l f T a. \<lbrakk>l \<in> dom f; lc (Obj f T); lc a\<rbrakk> \<Longrightarrow> P1 (Call (Obj f T) l a) (the (f l)\<^bsup>[Obj f T,a]\<^esup>)"   and "\<And>L t t'. \<lbrakk>finite L; \<forall>s p. s \<notin> L \<and> p \<notin> L \<and> s \<noteq> p \<longrightarrow> (\<exists>t''. t\<^bsup>[Fvar s,Fvar p]\<^esup> -> t'' \<and> P1 (t\<^bsup>[Fvar s,Fvar p]\<^esup>) t'' \<and> t' = \<sigma>[s,p] t'')\<rbrakk> \<Longrightarrow> P2 t t'" shows "P1 t t'"
subst_domain_subst_Fun_single: fixes x :: "'a"   and f :: "'b"   and T :: "('b, 'a) Term.term list" shows "subst_domain (Var(x := Fun f T)) = {x}"
nontrivial_relative_homology_group: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set" shows "relative_homology_group (int p) X S = relcycle_group p X S Mod singular_relboundary_set p X S"
P_mod_coeffTr: fixes I :: "'a set"   and d :: "'a" assumes "ideal S I"   and "d \<in> carrier S" shows "P_mod R S X I d = (d \<in> I)"
lexordp_conv_lexordp_eq: fixes xs :: "'a list"   and ys :: "'a list" shows "ord_class.lexordp xs ys = (lexordp_eq xs ys \<and> \<not> lexordp_eq ys xs)"
intro: fixes point_set :: "'a set"   and block_collection :: "'a set multiset"   and u_block_size :: "int"   and grouping :: "int"   and index :: "int" assumes "incomplete_design point_set block_collection u_block_size"   and "t_wise_balance point_set block_collection grouping index"   and "t_design_axioms u_block_size grouping" shows "t_design point_set block_collection u_block_size grouping index"
comp_reduce: fixes f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and g :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and k :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and h :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "f \<cdot> g = k"   and "seq f g"   and "seq g h" shows "f \<cdot> g \<cdot> h = k \<cdot> h"
Abs_poly_cases: fixes x :: "'b poly" obtains y :: "nat \<Rightarrow> 'b" where "x = Abs_poly y" and "y \<in> {f. \<forall>\<^sub>\<infinity>n. f n = (0::'b)}"
set_eqI: fixes A :: "'h set"   and B :: "'h set" assumes "\<And>x. (x \<in> A) = (x \<in> B)" shows "A = B"
PSQ: shows "(P, S, Q) = Smith_normal_form_of A bezout"
not_triv_relI: fixes a :: "'a"   and A :: "'a set"   and f :: "'a \<Rightarrow> int" assumes "a \<in> A"   and "f a \<noteq> 0" shows "f \<noteq> (\<lambda>_\<in>A. 0)"
consistent_def: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and i :: "nat"   and e :: "nat" shows "consistent \<psi> i e \<equiv> if \<forall>k<e_length e. \<psi> i k \<down> then if \<forall>k<e_length e. \<psi> i k \<down>= e_nth e k then Some 0 else Some 1 else None"
simps: fixes a1 :: "'a list"   and a2 :: "'b sequent" shows "upRulesp a1 a2 = ((\<exists>c R Fs ps. a1 = ps \<and> a2 = c \<and> c = ( \<LM> Compound R Fs  \<RM> \<Rightarrow>* Em) \<and> ps \<noteq> []) \<or> (\<exists>c F Fs ps. a1 = ps \<and> a2 = c \<and> c = ( \<Empt> \<Rightarrow>* Compound F Fs) \<and> ps \<noteq> []))"
embed_le_1: fixes G :: "'c \<Rightarrow> bool"   and s :: "'c" shows "\<guillemotleft> G \<guillemotright> s \<le> 1"
quasi_isometry_onD_3: fixes lambda :: "real"   and C :: "real"   and X :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "lambda C -quasi_isometry_on X f" shows "1 \<le> lambda"
d_IN_diff: fixes g :: "'a \<times> 'a \<Rightarrow> ennreal"   and y :: "'a"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "\<And>x. g (x, y) \<le> f (x, y)"   and "d_IN g y \<noteq> \<top>" shows "d_IN (\<lambda>e. f e - g e) y = d_IN f y - d_IN g y"
continuous_map_iff_continuous: fixes S :: "'b set"   and g :: "'b \<Rightarrow> 'c" shows "continuous_map (top_of_set S) euclidean g = continuous_on S g"
cases: fixes a :: "'b stream"   and A :: "'b set" assumes "a \<in> streams A" obtains aa :: "'b" and s :: "'b stream" where "a = aa ## s" and "aa \<in> A" and "s \<in> streams A"
ipassmt_nowild: shows "ipassmt_sanity_nowildcards ipassmt"
pmf_map: fixes f :: "'e \<Rightarrow> 'd"   and p :: "'e pmf"   and x :: "'d" shows "pmf (map_pmf f p) x = measure_pmf.prob p (f -` {x})"
substAbs_def: fixes xs :: "'varSort"   and X :: "('e, 'f, 'varSort, 'var, 'g) qTerm set"   and x :: "'var" shows "substAbs xs X x \<equiv> psubstAbs (idEnv [x \<leftarrow> X]_xs)"
le_approx_def: fixes P :: "'b process"   and Q :: "'b process" shows "P \<sqsubseteq> Q \<equiv> D Q \<subseteq> D P \<and> (\<forall>s. s \<notin> D P \<longrightarrow> Ra P s = Ra Q s) \<and> min_elems (D P) \<subseteq> T Q"
assms: shows "x \<in>\<^sub>\<circ> A \<inter>\<^sub>\<circ> B"
surjective: fixes r :: "('a, 'b) Graph_scheme" shows "r = \<lparr>verts = V\<^bsub>r\<^esub>, arcs = E\<^bsub>r\<^esub>, \<dots> = more r\<rparr>"
all_set_sing_pow: fixes w :: "'b list"   and a :: "'b" shows "(\<forall>b. b \<in> set w \<longrightarrow> b = a) = w \<in> [a]*"
analz_Un_implAuthSet: fixes G :: "msg set"   and H :: "msg set" assumes "G \<subseteq> payload"   and "Enc_keys_clean H" shows "analz ({implAuth A B M |A B M. M \<in> G} \<union> H) \<subseteq> synth (analz (G \<union> H)) \<union> - payload"
ring_im: fixes V :: "nat set set" shows "ring (im0.im_sheaf V) (im0.add_im_sheaf V) (im0.mult_im_sheaf V) (im0.zero_im_sheaf V) (im0.one_im_sheaf V)"
simps_2: fixes x :: "'a ltl"   and xs :: "'a ltl list" shows "Or (x # xs) = foldl LTLOr x xs"
step_def: fixes s :: "'a list"   and r :: "'a"   and a :: "nat \<times> nat list" shows "step s r a = (let (k, sws) = a in mtf2 k r (swaps sws s))"
finprod_cong': fixes A :: "'e set"   and B :: "'e set"   and g :: "'e \<Rightarrow> 'a"   and f :: "'e \<Rightarrow> 'a" assumes "A = B"   and "g \<in> B \<rightarrow> carrier G"   and "\<And>i. i \<in> B \<Longrightarrow> f i = g i" shows "finprod G f A = finprod G g B"
two_comparisons_into_case_order_12: fixes x :: "'a"   and y :: "'a"   and P :: "'d"   and Q :: "'d"   and R :: "'d" shows "(if x = y then P else if le x y then Q else R) = (case comp x y of Eq \<Rightarrow> P | Lt \<Rightarrow> Q | Gt \<Rightarrow> R)"
le_ASSERTI: fixes \<Phi> :: "bool"   and M :: "'a nres"   and M' :: "'a nres" assumes "\<Phi> \<Longrightarrow> M \<le> M'" shows "M \<le> assert \<Phi> \<bind> (\<lambda>_. M')"
psubstAll_ident: fixes P :: "('a, 'b, 'varSort, 'var, 'c) param"   and X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set" assumes "goodPar P" shows "(good X \<longrightarrow> (\<forall>rho\<in>envsOf P. (\<forall>zs z. freshEnv zs z rho \<or> fresh zs z X) \<longrightarrow> X #[rho] = X)) \<and> (goodAbs A \<longrightarrow> (\<forall>rho\<in>envsOf P. (\<forall>zs z. freshEnv zs z rho \<or> freshAbs zs z A) \<longrightarrow> A $[rho] = A))"
set_minus_filter_out: fixes xs :: "'b list"   and y :: "'b" shows "set xs - {y} = set (filter (\<lambda>x. x \<noteq> y) xs)"
br'_rcm_aux2: fixes Q :: "'d set"   and W :: "'d set"   and rcm :: "('d, 'e) ta_rule \<Rightarrow> nat option"   and \<delta> :: "('d, 'e) ta_rule set"   and q :: "'d" assumes "(Q, W, rcm) \<in> br'_invar \<delta>"   and "q \<in> W" shows "Q \<union> br_dsq \<delta> q (Q, W) = Q \<union> {lhs r |r. r \<in> \<delta> \<and> q \<in> set (rhsq r) \<and> the (rcm r) \<le> Suc 0}"
Pareto_strict_iff: fixes x :: "'alt"   and y :: "'alt" shows "x \<prec>[Pareto R] y = ((\<forall>i\<in>agents. R i x y) \<and> (\<exists>i\<in>agents. x \<prec>[R i] y))"
tabulation_axioms: shows "tabulation (\<cdot>) (\<star>) \<a> \<i> src trg r \<rho> f g"
distinct_11: fixes x2 :: "nat"   and x31 :: "nat"   and x32 :: "exp" shows "ENat x2 \<noteq> ELam x31 x32"
A20: shows "A2 = 0\<^sub>m n 1"
arrow_of_spans_of_maps_axioms: shows "arrow_of_spans_of_maps (\<cdot>) (\<star>) \<a> \<i> src trg (tab\<^sub>0 r) (tab\<^sub>1 r) (tab\<^sub>0 r) (tab\<^sub>1 r) r.chine"
mem\<^sub>A_of_def: fixes mem\<^sub>C :: "'Var \<Rightarrow> 'Val" shows "gen_refine.mem\<^sub>A_of mem\<^sub>C \<equiv> \<lambda>x\<^sub>A. mem\<^sub>C (id x\<^sub>A)"
AAss_\<tau>red1r_xt2: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and i :: "'addr expr1"   and xs :: "'addr val list"   and i' :: "'addr expr1"   and xs' :: "'addr val list"   and a :: "'addr val"   and e :: "'addr expr1" assumes "\<tau>red1gr uf P t h (i, xs) (i', xs')" shows "\<tau>red1gr uf P t h (Val a\<lfloor>i\<rceil> := e, xs) (Val a\<lfloor>i'\<rceil> := e, xs')"
simps: fixes a :: "'a stream" shows "run a = (\<exists>x y xs. a = x ## y ## xs \<and> E x y \<and> run (y ## xs))"
prob_space_distr: fixes f :: "('s \<Rightarrow> 's) \<Rightarrow> 'a"   and M' :: "'a measure" assumes "P.random_variable M' f" shows "prob_space (distr (Pi\<^sub>M UNIV (\<lambda>x. measure_pmf (K x))) M' f)"
mem_cone: fixes S :: "'c set"   and x :: "'c"   and c :: "real" assumes "cone S"   and "x \<in> S"   and "0 \<le> c" shows "c *\<^sub>R x \<in> S"
reduction_def: shows "reduction = (4 + \<alpha>) / (4 * \<alpha>)"
valid_path_linepath: fixes a :: "'a"   and b :: "'a" shows "valid_path (linepath a b)"
NDVDJ_l: fixes i :: "int"   and n :: "int"   and s :: "MIR.num" assumes "0 < i"   and "0 < n"   and "numbound0 s" shows "isrlfm (NDVDJ i n s)"
drop_take_drop_unsplit: fixes i :: "nat"   and j :: "nat"   and l :: "'c list" assumes "i \<le> j" shows "drop i (take j l) @ drop j l = drop i l"
eql_subst_trm_Iff: fixes t :: "trm"   and u :: "trm"   and i :: "name"   and trm :: "trm" shows "{t EQ u} \<turnstile> subst i t trm EQ subst i u trm"
assms_2: shows "\<lparr>s;vs;es\<rparr> \<leadsto>_ i \<lparr>s';vs';es'\<rparr>"
n_mult_lb2: fixes x :: "'a"   and y :: "'a" shows "n x \<cdot> n y \<sqsubseteq> n y"
local_typedef1: shows "\<exists>Rep Abs. type_definition Rep Abs {0..<int m}"
VAR: fixes s :: "'a" assumes "I s"   and "b s"   and "f s \<le> SPEC I" shows "f s \<le> SPEC (\<lambda>s'. (s', s) \<in> V)"
Int_greatest: fixes C :: "'a set"   and A :: "'a set"   and B :: "'a set" assumes "C \<subseteq> A"   and "C \<subseteq> B" shows "C \<subseteq> A \<inter> B"
fast_bv_to_nat_Nil: fixes k :: "num" shows "fast_bv_to_nat_helper [] k = k"
OclIsNew_def: fixes X :: "'d state \<times> 'd state \<Rightarrow> 'e" shows "X.oclIsNew() \<equiv> \<lambda>\<tau>. if (\<delta> X) \<tau> = true \<tau> then \<lfloor>\<lfloor>oid_of (X \<tau>) \<notin> dom (heap (fst \<tau>)) \<and> oid_of (X \<tau>) \<in> dom (heap (snd \<tau>))\<rfloor>\<rfloor> else invalid \<tau>"
ch2ch_cont: fixes f :: "'e \<Rightarrow> 'f"   and Y :: "nat \<Rightarrow> 'e" assumes "cont f"   and "chain Y" shows "chain (\<lambda>i. f (Y i))"
concat_map_singleton: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "concat (map (\<lambda>x. [f x]) xs) = map f xs"
double_induction_lemma: fixes s :: "dB"   and t :: "dB"   and r :: "dB"   and ss :: "dB list" assumes "termip (\<rightarrow>\<^sub>\<beta>) s"   and "termip (\<rightarrow>\<^sub>\<beta>) t"   and "t = r[s/0] \<degree>\<degree> ss" shows "termip (\<rightarrow>\<^sub>\<beta>) (Abs r \<degree> s \<degree>\<degree> ss)"
case_1: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'a" shows "(case None of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = f1"
delta_pf_def: fixes N :: "bool stream measure"   and der :: "bool stream \<Rightarrow> real"   and matur :: "nat" shows "delta_pf N der matur = qty_single stk (delta_predict N der matur)"
n_gt: fixes F :: "nat \<Rightarrow> 'b poly"   and n :: "nat \<Rightarrow> nat"   and \<delta> :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'b"   and k :: "nat"   and \<beta> :: "nat \<Rightarrow> 'b"   and i :: "nat" assumes "primitive_remainder_sequence F n \<delta> f k \<beta>"   and "2 \<le> i"   and "i < k" shows "n (Suc i) < n i"
compactness_L: fixes A :: "'a set"   and \<phi> :: "'a" assumes "A \<turnstile> \<phi>" shows "\<exists>A'\<subseteq>A. finite A' \<and> A' \<turnstile> \<phi>"
antisym: fixes b :: "'b multiset"   and a :: "'b multiset" assumes "b \<subseteq># a"   and "a \<subseteq># b" shows "a = b"
iteratei_rule_insert_P: fixes m :: "'s"   and I :: "'u set \<Rightarrow> '\<sigma> \<Rightarrow> bool"   and \<sigma>0 :: "'\<sigma>"   and c :: "'\<sigma> \<Rightarrow> bool"   and f :: "'u \<times> 'v \<Rightarrow> '\<sigma> \<Rightarrow> '\<sigma>"   and P :: "'\<sigma> \<Rightarrow> bool" assumes "invar m"   and "I {} \<sigma>0"   and "\<And>k v it \<sigma>. \<lbrakk>c \<sigma>; k \<in> dom (\<alpha> m) - it; \<alpha> m k = Some v; it \<subseteq> dom (\<alpha> m); I it \<sigma>\<rbrakk> \<Longrightarrow> I (insert k it) (f (k, v) \<sigma>)"   and "\<And>\<sigma>. I (dom (\<alpha> m)) \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> it. \<lbrakk>it \<subseteq> dom (\<alpha> m); it \<noteq> dom (\<alpha> m); \<not> c \<sigma>; I it \<sigma>\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (iteratei m c f \<sigma>0)"
assms_2: shows "\<guillemotleft>g : a' \<rightarrow>\<^sub>C a\<guillemotright>"
Max_singleton: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "class.linorder less_eq less" shows "linorder.Max less_eq {x} = x"
analytic_at: fixes f :: "complex \<Rightarrow> complex"   and z :: "complex" shows "(f analytic_on {z}) = (\<exists>s. open s \<and> z \<in> s \<and> f holomorphic_on s)"
resultant_zero_imp_common_factor: fixes p :: "'a poly"   and q :: "'a poly" assumes "0 < degree p \<or> 0 < degree q"   and "resultant p q = (0::'a)" shows "\<not> comm_monoid_mult_class.coprime p q"
Abs_lifted_pair_inverse: fixes y :: "('c, 'd) phi \<Rightarrow> 'c \<times> 'd Filters.filter" assumes "y \<in> {pf. \<forall>f. pf f \<in> triple.pairs (Rep_phi f)}" shows "Rep_lifted_pair (Abs_lifted_pair y) = y"
dg_Rel_def: fixes \<alpha> :: "V" shows "dg_Rel \<alpha> = [Vset \<alpha>, ZFC_in_HOL.set {T. arr_Rel \<alpha> T}, \<lambda>T\<in>\<^sub>\<circ>ZFC_in_HOL.set {T. arr_Rel \<alpha> T}. T\<lparr>ArrDom\<rparr>, \<lambda>T\<in>\<^sub>\<circ>ZFC_in_HOL.set {T. arr_Rel \<alpha> T}. T\<lparr>ArrCod\<rparr>]\<^sub>\<circ>"
filter_filter: fixes P :: "'d \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool"   and xs :: "'d list" shows "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
blocksFinal: fixes wf_md :: "(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 \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and Vs :: "char list list"   and Ts :: "ty list"   and vs :: "val list"   and e :: "expr"   and vs' :: "val list"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l :: "char list \<Rightarrow> val option"   and E :: "char list \<Rightarrow> ty option" assumes "wf_prog wf_md P"   and "length Vs = length Ts"   and "length vs = length Ts"   and "final e"   and "P \<turnstile> Ts Casts vs to vs' " shows "P,E \<turnstile> \<langle>blocks (Vs, Ts, vs, e),(h, l)\<rangle> \<rightarrow>* \<langle>e,(h, l)\<rangle>"
prj_in_hom_5: shows "\<guillemotleft>doms.Prj\<^sub>1\<^sub>0 : Chn (local.dom \<mu> \<star> local.dom \<nu> \<star> local.dom \<pi>) \<rightarrow>\<^sub>C doms.\<nu>.apex\<guillemotright>"
set_val_locs_impl: shows "set_val_locs = CD.a_set_val_locs"
intros: fixes p :: "'d \<times> 'a" shows "snd p \<in> Basic_BNFs.snds p"
C_eq_compile: fixes p :: "((int \<times> int) set set, int) Combinators"   and l :: "(int \<times> int) set set set list" assumes "DenyAll \<in> set (policy2list p)"   and "all_in_list (policy2list p) l"   and "allNetsDistinct (policy2list p)" shows "C (list2FWpolicy (removeAllDuplicates (insertDenies (separate (FWNormalisationCore.sort (removeShadowRules2 (remdups (rm_MT_rules C (insertDeny (removeShadowRules1 (policy2list p)))))) l))))) = C p"
left_one_impl: fixes a :: "'a" shows "(1::'a) r\<rightarrow> a = a"
eucl_rel_poly_add_left: fixes x :: "'b poly"   and y :: "'b poly"   and q :: "'b poly"   and r :: "'b poly"   and x' :: "'b poly"   and q' :: "'b poly"   and r' :: "'b poly" assumes "eucl_rel_poly x y (q, r)"   and "eucl_rel_poly x' y (q', r')" shows "eucl_rel_poly (x + x') y (q + q', r + r')"
pmf_map_inj': fixes f :: "'e \<Rightarrow> 'f"   and M :: "'e pmf"   and x :: "'e" assumes "inj f" shows "pmf (map_pmf f M) (f x) = pmf M x"
pix_not_None: shows "inv_map \<pi> x \<noteq> None"
simps_1: fixes A :: "'a mat" shows "A ^\<^sub>m 0 = 1\<^sub>m (dim_row A)"
l_first_in_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_first_in_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M to_tree_order first_in_tree_order"
iterator_to_ordered_iterator_mergesort_def: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and it :: "('d list \<Rightarrow> bool) \<Rightarrow> ('d \<Rightarrow> 'd list \<Rightarrow> 'd list) \<Rightarrow> 'd list \<Rightarrow> 'd list" shows "iterator_to_ordered_iterator_mergesort R it = iterator_to_ordered_iterator (mergesort_by_rel R) it"
edges_ins_edge: fixes e :: "'b \<times> 'b"   and g :: "'b ugraph" shows "edges (ins_edge e g) = (if fst e = snd e then edges g else {e, prod.swap e} \<union> edges g)"
subgroup_nonempty: fixes G :: "('c, 'd) monoid_scheme" shows "\<not> subgroup {} G"
pmf_of_set_singleton: fixes x :: "'b" shows "pmf_of_set {x} = return_pmf x"
ord_def: fixes x :: "'a" shows "ord x \<equiv> SOME d. \<forall>n. (x [^] n = \<one>) = (d dvd n)"
inj_on_add: fixes a :: "'a"   and A :: "'a set" shows "inj_on ((+) a) A"
wlconf_cong: fixes G :: "prog"   and x :: "lname \<Rightarrow> val option"   and s :: "st"   and l :: "'a \<Rightarrow> val option"   and L :: "'a \<Rightarrow> ty option" shows "G,set_locals x s\<turnstile>l[\<sim>\<Colon>\<preceq>]L = G,s\<turnstile>l[\<sim>\<Colon>\<preceq>]L"
BB_ap: fixes f :: "('c, 'e \<Rightarrow> 'd) state"   and x :: "('c, 'e) state" shows "B (B f) \<diamondop> B (B x) = B (B (f \<diamondop> x))"
that: fixes g :: "complex \<Rightarrow> complex" assumes "g holomorphic_on S"   and "\<And>z. z \<in> S \<Longrightarrow> f z = (g z)\<^sup>2" shows "thesis"
simps: fixes rx :: "real interval"   and ix :: "real interval"   and ry :: "real interval"   and iy :: "real interval" shows "Complex_Interval rx ix + Complex_Interval ry iy = Complex_Interval (rx + ry) (ix + iy)"
assms: shows "x \<in> space (Pi\<^sub>M J (\<lambda>_. M))"
u_finite: fixes y :: "'v" shows "u y \<noteq> \<top>"
contra: shows "\<not> (v \<in> VV p \<and> \<sigma> v \<noteq> w)"
prim_red_cong1: fixes e1 :: "exp"   and e1' :: "exp"   and f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and e2 :: "exp" assumes "e1 \<longrightarrow>* e1'" shows "EPrim f e1 e2 \<longrightarrow>* EPrim f e1' e2"
frameChainFresh_1: fixes xvec :: "name list"   and \<Psi> :: "'b" shows "xvec \<sharp>* (\<langle>\<epsilon>, \<Psi>\<rangle>) = xvec \<sharp>* \<Psi>"
msb_min_neg: fixes w :: "32 Word.word" assumes "msb (- w)"   and "msb w" shows "uint w = 2 ^ LENGTH(31)"
arr_eqI: fixes t :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and t' :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "arr t"   and "arr t'"   and "Dom t = Dom t'"   and "Cod t = Cod t'"   and "Map t = Map t'" shows "t = t'"
B_def: shows "B \<equiv> {x. poly q x = 0 \<and> a < x \<and> x < b}"
at_within_def: fixes "open" :: "'a set \<Rightarrow> bool"   and a :: "'a"   and s :: "'a set" assumes "class.topological_space open" shows "topological_space.at_within open a s = inf (topological_space.nhds open a) (principal (s - {a}))"
fmdrop_empty: fixes a :: "'a" shows "fmdrop a fmempty = fmempty"
fml_sem_implies: fixes I :: "interp"   and \<phi> :: "fml"   and \<psi> :: "fml" shows "fml_sem I (\<phi> \<rightarrow> \<psi>) = - fml_sem I \<phi> \<union> fml_sem I \<psi>"
bin_sign_simps_2: shows "bin_sign 1 = 0"
mult_strict_left_mono: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less a b"   and "less zero c" shows "less (times c a) (times c b)"
cospan_\<nu>\<pi>: shows "C.cospan \<pi>.leg0 \<rho>.leg1"
singular_relcycle_chain_map: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int"   and X' :: "'c topology"   and g :: "'b \<Rightarrow> 'c"   and T :: "'c set" assumes "singular_relcycle p X S c"   and "continuous_map X X' g"   and "g ` S \<subseteq> T" shows "singular_relcycle p X' T (chain_map p g c)"
size_bounded: fixes t :: "'b bro"   and h :: "nat" assumes "t \<in> B h" shows "size (fib_tree h) \<le> size t"