hcju/isabelleps huggingface dataset:
abs_less_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (abs a) b = (less a b \<and> less (uminus a) b)"
edges_less_eq_def: fixes a :: "'a \<times> 'b \<times> 'a"   and b :: "'a \<times> 'b \<times> 'a" shows "edges_less_eq a b \<equiv> fst (snd a) \<le> fst (snd b)"
bgf: fixes i :: "'a \<times> real" assumes "i \<in> Basis" shows "basg (basf i) = i"
mat_of_cblinfun_vector_to_cblinfun: fixes \<psi> :: "'c" shows "mat_of_cblinfun (vector_to_cblinfun \<psi>) = mat_of_cols (length canonical_basis) [vec_of_basis_enum \<psi>]"
abs_eq: fixes x :: "32 word" shows "uint16_of_uint32 (Abs_uint32 x) = Abs_uint16 (ucast x)"
eq_mrexp2_eq: fixes r :: "'b mrexp2"   and s :: "'b mrexp2" assumes "ok2 r"   and "ok2 s" shows "eq_mrexp2 r s = (r = s)"
IH: fixes Pi :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) set"   and s :: "('a, 'b) fmap" assumes "as \<in> valid_plans Pi"   and "s \<in> valid_states Pi" shows "valid_path Pi (statelist' s as)"
one_poly_def: shows "one_poly \<equiv> [(1, 1::'c)]"
bind_mono: fixes B :: "('e \<Rightarrow> 'f option) \<Rightarrow> 'g option"   and C :: "'g \<Rightarrow> ('e \<Rightarrow> 'f option) \<Rightarrow> 'h option" assumes "monotone option.le_fun option_ord B"   and "\<And>y. monotone option.le_fun option_ord (C y)" shows "monotone option.le_fun option_ord (\<lambda>f. B f \<bind> (\<lambda>y. C y f))"
simps: fixes rha :: "'e \<Rightarrow> 'f"   and boa :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and rhb :: "'g \<Rightarrow> 'h"   and bob :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and a1 :: "('e, 'f, 'g, 'h) G\<^sub>m"   and a2 :: "('e, 'f, 'g, 'h) G\<^sub>m" shows "blinding_of_G rha boa rhb bob a1 a2 = (\<exists>x y. a1 = G\<^sub>m x \<and> a2 = G\<^sub>m y \<and> blinding_of_F (root_hash_T rha) (blinding_of_T rha boa) rhb bob x y)"
set_arrow_def: fixes U :: "'b set"   and f :: "'b set_arrow" shows "set_arrow U f = (set_dom f \<subseteq> U \<and> set_cod f \<subseteq> U \<and> set_func f \<in> set_dom f \<rightarrow> set_cod f \<and> set_func f \<in> extensional (set_dom f))"
Jmp: fixes G :: "prog"   and s :: "st"   and j :: "jump" shows "G\<turnstile>Norm s \<midarrow>Jmp j\<rightarrow> (Some (Jump j), s)"
ContractsWithBilateralSubstitutes_axioms: shows "ContractsWithBilateralSubstitutes Xd Xh Pd Ch"
69: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> x \<sqinter> - - y = x"
underS_E: fixes i :: "'d"   and R :: "('d \<times> 'd) set"   and j :: "'d" assumes "i \<in> underS R j" shows "i \<noteq> j \<and> (i, j) \<in> R"
escape_rate_def: fixes x :: "'a" shows "escape_rate x = integral\<^sup>L (count_space UNIV) (R x)"
\<eta>s0_def: shows "\<eta>s0 = map \<eta>s0f [0..<n]"
lit_ord_neg_lit_lhs: fixes x1 :: "'a literal"   and s :: "'a trm"   and t1 :: "'a trm"   and x2 :: "'a literal"   and t2 :: "'a trm" assumes "orient_lit x1 s t1 pos"   and "orient_lit x2 s t2 neg"   and "vars_of_lit x1 = {}"   and "vars_of_lit x2 = {}" shows "(x1, x2) \<in> lit_ord"
chain_dual: shows "Complete_Partial_Order.chain (\<lambda>x y. y \<le> x) = Complete_Partial_Order.chain (\<le>)"
group_add_ow_axioms_def: fixes U :: "'a set"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a" shows "group_add_ow_axioms U minus plus zero uminus \<equiv> (\<forall>a. a \<in> U \<longrightarrow> plus (uminus a) a = zero) \<and> (\<forall>a b. a \<in> U \<longrightarrow> b \<in> U \<longrightarrow> plus a (uminus b) = minus a b)"
E_TAbs: fixes T\<^sub>1\<^sub>1 :: "POPLmark.type"   and t\<^sub>1\<^sub>2 :: "trm"   and T\<^sub>2 :: "POPLmark.type" shows "(\<lambda><:T\<^sub>1\<^sub>1. t\<^sub>1\<^sub>2) \<bullet>\<^sub>\<tau> T\<^sub>2 \<longmapsto> t\<^sub>1\<^sub>2[0 \<mapsto>\<^sub>\<tau> T\<^sub>2]"
diff_add_0: fixes n :: "nat"   and m :: "nat" shows "n - (n + m) = 0"
graph_def: fixes Obj :: "'a set"   and Arr :: "'b set"   and Dom :: "'b \<Rightarrow> 'a"   and Cod :: "'b \<Rightarrow> 'a" shows "graph Obj Arr Dom Cod \<equiv> (\<forall>x. x \<in> Arr \<longrightarrow> Dom x \<in> Obj) \<and> (\<forall>x. x \<in> Arr \<longrightarrow> Cod x \<in> Obj)"
normalfilter_def: fixes R :: "'b \<Rightarrow> 'a \<Rightarrow> bool" shows "normalfilter R = {a. R a (1::'a)}"
rel_spmf_simps: fixes R :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'b spmf"   and q :: "'d spmf" shows "rel_spmf R p q = (\<exists>pq. (\<forall>(x, y)\<in>set_spmf pq. R x y) \<and> map_spmf fst pq = p \<and> map_spmf snd pq = q)"
nth_least_strict_mono: fixes l :: "nat"   and A :: "'b set"   and k :: "nat" assumes "enat l < esize A"   and "k < l" shows "nth_least A k < nth_least A l"
F_axioms_6: fixes AA :: "('d, 'e, 'f) category_scheme"   and BB :: "('g, 'h, 'i) category_scheme"   and F :: "('d, 'e, 'g, 'h) functor" assumes "Functor F : AA \<longrightarrow> BB" shows "\<forall>f\<in>Ar\<^bsub>AA\<^esub>. \<forall>g\<in>Ar\<^bsub>AA\<^esub>. Cod\<^bsub>AA\<^esub> f = Dom\<^bsub>AA\<^esub> g \<longrightarrow> F \<^bsub>\<a>\<^esub> (g \<bullet>\<^bsub>AA\<^esub> f) = F \<^bsub>\<a>\<^esub> g \<bullet>\<^bsub>BB\<^esub> F \<^bsub>\<a>\<^esub> f"
module_homTr: fixes N :: "('g, 'b, 'h) Module_scheme"   and f :: "'a \<Rightarrow> 'g" assumes "R module N"   and "f \<in> mHom R M N" shows "f \<in> mHom R M (mimg\<^bsub>R M,N\<^esub> f)"
homeomorphic_ball_UNIV: fixes r :: "real"   and a :: "'b" assumes "0 < r" shows "ball a r homeomorphic UNIV"
JN_empty: fixes F :: "'d \<Rightarrow> 'c program" shows "JOIN {} F = \<bottom>"
less_add_same_cancel2: fixes a :: "'k multiset"   and b :: "'k multiset" shows "(a \<subset># b + a) = ({#} \<subset># b)"
range_binary_eq: fixes a :: "'c"   and b :: "'c" shows "range (binary a b) = {a, b}"
at_top_within_at_top_bounded_right': fixes a :: "'b"   and b :: "'b" assumes "a < b" shows "at_top_within {..<b} = at_left b"
sel_specD2: fixes sel :: "(('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> ((('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> nat \<times> 'h \<Rightarrow> ((('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and ps :: "((('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and gs :: "(('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and bs :: "(('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and data :: "nat \<times> 'h" assumes "sel_spec sel"   and "ps \<noteq> []" shows "set (sel gs bs ps data) \<subseteq> set ps"
weak_group_morphism_axioms_def: fixes f :: "'d \<Rightarrow> 'e"   and H :: "'d set"   and G :: "('d, 'f) monoid_scheme" shows "weak_group_morphism_axioms f H G \<equiv> \<forall>a b. a \<in> carrier G \<longrightarrow> b \<in> carrier G \<longrightarrow> (f a = f b) = (a \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> b \<in> H)"
euclidean_representation: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and Basis :: "'c set"   and x :: "'c" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis" shows "comm_monoid_add.sum plus zero (\<lambda>b. scaleR (inner x b) b) Basis = x"
red_diff_rtrancl_cs: fixes F :: "('a \<Rightarrow>\<^sub>0 'c) set"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c" assumes "(punit.red F)\<^sup>*\<^sup>* (p - q) 0" shows "relation.cs (punit.red F) p q"
ValTT_below_iff: fixes x :: "ValD" shows "(ValTT \<sqsubseteq> x) = (x = ValTT)"
p'_1: shows "lprefix (llist_of p') (induced_play \<sigma> p)"
simps: fixes x :: "name"   and p :: "tm"   and q :: "tm"   and v :: "name"   and y :: "name"   and x' :: "name" assumes "atom x \<sharp> (p, q, v, y, x')"   and "atom x' \<sharp> (p, q, v, y)"   and "atom y \<sharp> (p, q, v)"   and "atom v \<sharp> (p, q)" shows "ExistsP p q = SyntaxN.Ex x (SyntaxN.Ex x' (SyntaxN.Ex y (SyntaxN.Ex v (FormP (Var x) AND VarNonOccFormP (Var v) (Var y) AND AbstFormP (Var v) Zero (Var x) (Var x') AND p EQ Q_Imp (Var x) (Var y) AND q EQ Q_Imp (Q_Ex (Var x')) (Var y)))))"
abs_not_less_zero: fixes abs :: "'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "\<not> less (abs a) zero"
op_list_singleton_def: fixes x :: "'b" shows "op_list_singleton x \<equiv> [x]"
step: fixes xs :: "'a llist"   and x' :: "'a" assumes "x \<in> lset xs"   and "x \<noteq> x'"   and "P xs" shows "P (LCons x' xs)"
covimi: shows "ov\<inverse> O m\<inverse> \<subseteq> b\<inverse>"
Stream_image: fixes x :: "'b"   and y :: "'b stream"   and x' :: "'b"   and Y :: "'b stream set" shows "(x ## y \<in> (##) x' ` Y) = (x = x' \<and> y \<in> Y)"
linear_gf: shows "linear (*s) (*s) (g \<circ> f)"
lcoset_eq_reps: fixes a :: "'b"   and A :: "'b set"   and B :: "'b set" assumes "a +o A = a +o B" shows "A = B"
SV_inv2_inductive_1: shows "init sv_TS \<subseteq> Vinv2"
bind_return_pmf': fixes N :: "'e pmf" shows "N \<bind> return_pmf = N"
i_set_cut_le_ge_closed_disj: fixes I :: "nat set"   and t :: "nat"   and cut_op :: "nat set \<Rightarrow> nat \<Rightarrow> nat set" assumes "I \<in> i_set"   and "t \<in> I"   and "cut_op = (\<down>\<le>) \<or> cut_op = (\<down>\<ge>)" shows "cut_op I t \<in> i_set"
rcos_inv: fixes x :: "'a" assumes "x \<in> carrier G" shows "set_inv (H #> x) = H #> inv x"
locally_compact_homeomorphic_closed: fixes S :: "'a set" assumes "locally compact S"   and "DIM('a) < DIM('b)" obtains T :: "'b set" where "closed T" and "S homeomorphic T"
aut_dfa_def: fixes A :: "nat bdd list \<times> bool list"   and n :: "nat" shows "aut_dfa A n \<equiv> wf_dfa A n"
Throw\<^sub>1: fixes P :: "nat exp prog"   and e :: "nat exp"   and s\<^sub>0 :: "heap \<times> val list \<times> sheap"   and a :: "nat"   and s\<^sub>1 :: "heap \<times> val list \<times> sheap" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>addr a,s\<^sub>1\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>throw e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Throw a,s\<^sub>1\<rangle>"
SFAccNone: fixes P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and F :: "char list"   and D :: "char list"   and s :: "heap \<times> sfields \<times> sheap" assumes "\<nexists>b t. P \<turnstile> C has F,b:t in D" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sF{D},s\<rangle> \<Rightarrow> \<langle>THROW NoSuchFieldError,s\<rangle>"
power_mult_inverse_distrib: fixes x :: "'b"   and m :: "nat" shows "x ^ m * inverse x = inverse x * x ^ m"
left_lang_def: fixes q :: "hf" shows "MN.left_lang q \<equiv> {u. MN.nextl (dfa.init Canon.DFA) u = q}"
less_imp_le: fixes x :: "'c list"   and y :: "'c list" assumes "strict_prefix x y" shows "prefix x y"
Polys_closed_hom_deg_set: fixes A :: "(('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set"   and X :: "'b set"   and z :: "nat" assumes "A \<subseteq> P[X]" shows "hom_deg_set z A \<subseteq> P[X]"
bw_diff_plus_pow: fixes k :: "nat"   and f :: "'c \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'c" shows "(bw_diff ^^ k) (\<lambda>x. f x + g x) = (\<lambda>x. (bw_diff ^^ k) f x + (bw_diff ^^ k) g x)"
inf_bot_right: fixes x :: "'a upred" shows "(x \<and> false) = false"
M_def: shows "M = mult_row A i (ass (A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'b))) div A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'b)))"
wqo_axioms_def: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "class.wqo_axioms less_eq \<equiv> \<forall>f. good less_eq f"
iMODb_eq_conv: fixes m :: "nat"   and c :: "nat"   and r :: "nat"   and r' :: "nat"   and m' :: "nat"   and c' :: "nat" assumes "0 < m"   and "0 < c" shows "([ r, mod m, c ] = [ r', mod m', c' ]) = (r = r' \<and> m = m' \<and> c = c')"
neg_le_0_iff_le: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq (uminus a) zero = less_eq zero a"
poly_mapping_neqE: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "p \<noteq> q" obtains v :: "'t" where "v \<in> keys p \<union> keys q" and "lookup p v \<noteq> lookup q v" and "\<And>u. lt_of_nat_term_order cmp_term v u \<Longrightarrow> lookup p u = lookup q u"
AS_def: shows "AS = {As \<in> lists all_AA. length As \<le> max_ary}"
ex_ivl_eq: fixes f :: "'b \<Rightarrow> 'b"   and T :: "real set"   and S :: "'b set"   and \<phi> :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and s :: "'b" assumes "local_flow f T S \<phi>"   and "s \<in> S" shows "ll_on_open.existence_ivl T (\<lambda>t. f) S 0 s = T"
cancel_imp_equal: fixes G :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and u :: "'c"   and v :: "'c" assumes "Group_Theory.group G composition unit"   and "composition u (monoid.inverse G composition unit v) = unit"   and "u \<in> G"   and "v \<in> G" shows "u = v"
Bossy'_CD_on_Cd: fixes X :: "'a set" shows "Bossy''.CD_on ds X = Bossy''.Cd (Xd d1h1) X \<union> Bossy''.Cd (Xd d3h1) X"
Fvars_tm_simps_2: fixes a :: "name" shows "Fvars (Var a) = {a}"
lconf_def: fixes typeof_addr :: "'a \<Rightarrow> 'b \<Rightarrow> htype option"   and P :: "'c prog"   and h :: "'a"   and l :: "String.literal \<Rightarrow> 'b val option"   and E :: "String.literal \<Rightarrow> ty option" shows "heap_base.lconf typeof_addr P h l E \<equiv> \<forall>V v. l V = \<lfloor>v\<rfloor> \<longrightarrow> (\<exists>T. E V = \<lfloor>T\<rfloor> \<and> heap_base.conf typeof_addr P h v T)"
UNION_empty_conv_2: fixes B :: "'d \<Rightarrow> 'c set"   and A :: "'d set" shows "(\<Union> (B ` A) = {}) = (\<forall>x\<in>A. B x = {})"
extf_trans: fixes zs :: "('s, 'v) tm list"   and A :: "('s, 'v) tm set"   and ys :: "('s, 'v) tm list"   and xs :: "('s, 'v) tm list"   and gt :: "('s, 'v) tm \<Rightarrow> ('s, 'v) tm \<Rightarrow> bool"   and f :: "'s" assumes "zs \<in> lists A"   and "ys \<in> lists A"   and "xs \<in> lists A"   and "\<forall>z\<in>A. \<forall>y\<in>A. \<forall>x\<in>A. gt z y \<longrightarrow> gt y x \<longrightarrow> gt z x"   and "extf f gt zs ys"   and "extf f gt ys xs" shows "extf f gt zs xs"
rep_eq: fixes x :: "'d \<Rightarrow> 'c \<Rightarrow>\<^sub>L real" shows "blinfun_apply (blinfun_vec x) = (\<lambda>xa. \<chi>i. blinfun_apply (x i) xa)"
C2: fixes \<Psi>' :: "'b"   and S :: "('a, 'b, 'c) psi"   and T :: "('a, 'b, 'c) psi"   and xvec :: "name list" assumes "(\<Psi>', S, T) \<in> Rel'"   and "xvec \<sharp>* \<Psi>'" shows "(\<Psi>', \<lparr>\<nu>*xvec\<rparr>S, \<lparr>\<nu>*xvec\<rparr>T) \<in> Rel'"
interaction_bound_def: fixes "consider" :: "'d \<Rightarrow> bool" shows "interaction_bound consider \<equiv> lfp.fixp_fun (\<lambda>interaction_bound gpv. \<Squnion>generat\<in>set_spmf (the_gpv gpv). case generat of Pure x \<Rightarrow> 0 | IO out c \<Rightarrow> if consider out then eSuc (\<Squnion>input. interaction_bound (c input)) else \<Squnion>input. interaction_bound (c input))"
UN_upper: fixes a :: "'f"   and A :: "'f set"   and B :: "'f \<Rightarrow> 'g set" assumes "a \<in> A" shows "B a \<subseteq> \<Union> (B ` A)"
primitive_extractor_correct_7: fixes m :: "'b match_expr"   and disc :: "'b \<Rightarrow> bool"   and sel :: "'b \<Rightarrow> 'c"   and C :: "'c \<Rightarrow> 'b"   and as :: "'c negation_type list"   and ms :: "'b match_expr" assumes "normalized_nnf_match m"   and "wf_disc_sel (disc, sel) C"   and "primitive_extractor (disc, sel) m = (as, ms)" shows "(\<not> has_disc disc m) = (as = [] \<and> ms = m)"
finite_UN_subsets: fixes A :: "'b set"   and B :: "'b \<Rightarrow> 'c set"   and b :: "'c set" assumes "\<forall>a\<in>A. B a \<subseteq> b"   and "finite b" shows "finite (\<Union> (B ` A))"
map_of_Cons_code_2: fixes l :: "'l"   and v :: "'k"   and ps :: "('l \<times> 'k) list"   and k :: "'l" shows "map_of ((l, v) # ps) k = (if l = k then Some v else map_of ps k)"
fin_string_enum_all_def: shows "fin_string_enum_all = (\<lambda>f. list_all f vals)"
homeomorphic_space_iff_homeomorphic: fixes S :: "'a set"   and T :: "'b set" shows "(top_of_set S homeomorphic_space top_of_set T) = S homeomorphic T"
null: shows "lnull (ltl (ldropWhile (Not \<circ> P) (ltake (enat (Suc (lfind P lst))) lst)))"
assms_3: shows "A \<in>\<^sub>\<circ> Vset \<alpha>"
set_of_appr_dropD: fixes xs :: "real list"   and n :: "nat"   and XS :: "'b list" assumes "xs \<in> set_of_appr (drop n XS)" shows "xs \<in> drop n ` set_of_appr XS"
param_opt_def: fixes i :: "'d ifex"   and t :: "'d ifex"   and e :: "'d ifex" shows "param_opt i t e = (if i = Trueif then Some t else if i = Falseif then Some e else if t = Trueif \<and> e = Falseif then Some i else if t = e then Some t else if e = Trueif \<and> i = t then Some Trueif else if t = Falseif \<and> i = e then Some Falseif else None)"
inner_prod_distrib_right: fixes u :: "'a vec"   and n :: "nat"   and v :: "'a vec"   and w :: "'a vec" assumes "u \<in> carrier_vec n"   and "v \<in> carrier_vec n"   and "w \<in> carrier_vec n" shows "inner_prod u (v + w) = inner_prod u v + inner_prod u w"
simps_8: fixes x :: "octo"   and y :: "octo" shows "Im7 (x + y) = Im7 x + Im7 y"
paths_intersect: fixes t :: "'b ttree"   and t' :: "'b ttree" shows "paths (t \<inter>\<inter> t') = paths t \<inter> paths t'"
lower_on_cl: fixes \<sigma> :: "('a \<times> 'a trm) list"   and \<eta> :: "('a \<times> 'a trm) list"   and C :: "'a eclause"   and x :: "'a" assumes "lower_on \<sigma> \<eta> (vars_of_cl (cl_ecl C))"   and "(Var x \<lhd> \<sigma>, Var x \<lhd> \<eta>) \<in> trm_ord"   and "x \<in> vars_of_cl (cl_ecl C)"   and "finite (cl_ecl C)" shows "((C, \<sigma>), C, \<eta>) \<in> ecl_ord"
trivial_imp_cyclic_group: fixes G :: "('a, 'b) monoid_scheme" assumes "trivial_group G" shows "cyclic_group G"
simps_111: fixes x131 :: "expr"   and x132 :: "expr"   and x133 :: "expr"   and x4 :: "val" shows "if (x131) x132 else x133 \<noteq> Val x4"
elims: fixes x :: "'d decision decision"   and y :: "'d decision" assumes "flat_1 x = y" obtains ya :: "'d" where "x = allow (allow ya__)" and "y = allow ya__"   | ya :: "'d" where "x = allow (deny yaa__)" and "y = allow yaa__"   | ya :: "'d" where "x = deny (allow yab__)" and "y = deny yab__"   | ya :: "'d" where "x = deny (deny ya)" and "y = deny ya"
sc_l_distr: fixes a :: "'b"   and b :: "'b"   and m :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "m \<in> carrier M" shows "(a \<plusminus>\<^bsub>R\<^esub> b) \<cdot>\<^sub>s m = a \<cdot>\<^sub>s m \<plusminus> b \<cdot>\<^sub>s m"
equal_mod_timpls_refl: fixes TI :: "('e set \<times> 'e set) list"   and t :: "(('f, 'g, 'e) prot_fun, 'h) Term.term" shows "equal_mod_timpls TI t t"
recvmsg_simps_1: fixes P :: "'a \<Rightarrow> bool"   and m :: "'a" shows "recvmsg P (broadcast m) = True"
wls_templateInduct: fixes rel :: "'sort \<Rightarrow> (('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<times> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set) set"   and phi :: "'sort \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> bool"   and phiAbs :: "'varSort \<times> 'sort \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> bool"   and s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and xs :: "'varSort"   and s' :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set" assumes "\<And>s X Y. \<lbrakk>wls s X; (X, Y) \<in> rel s\<rbrakk> \<Longrightarrow> wls s Y \<and> skel Y = skel X"   and "\<And>xs x. phi (asSort xs) (Var xs x)"   and "\<And>delta inp binp. \<lbrakk>wlsInp delta inp; wlsBinp delta binp; liftAll2 phi (arOf delta) inp; liftAll2 phiAbs (barOf delta) binp\<rbrakk> \<Longrightarrow> phi (stOf delta) (Op delta inp binp)"   and "\<And>s xs x X. \<lbrakk>isInBar (xs, s); wls s X; \<And>Y. (X, Y) \<in> rel s \<Longrightarrow> phi s Y\<rbrakk> \<Longrightarrow> phiAbs (xs, s) (Abs xs x X)" shows "(wls s X \<longrightarrow> phi s X) \<and> (wlsAbs (xs, s') A \<longrightarrow> phiAbs (xs, s') A)"
that_3: assumes "\<^bold>|\<rho> u\<^bold>| < \<^bold>|u\<^bold>|" shows "thesis"
l_set_tag_name_get_shadow_root\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and set_tag_name :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_tag_name_get_shadow_root\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_tag_name set_tag_name_locs get_shadow_root get_shadow_root_locs \<equiv> l_set_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_tag_name set_tag_name_locs \<and> l_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_shadow_root get_shadow_root_locs"
count_inI: fixes M :: "'k multiset"   and x :: "'k" assumes "count M x = 0 \<Longrightarrow> False" shows "x \<in># M"
periodic_set_subset: fixes S :: "real set"   and \<delta> :: "real"   and T :: "real set" assumes "periodic_set S \<delta>"   and "T \<subseteq> S" shows "periodic_set T \<delta>"
cccFix_cong: fixes heap2 :: "(Vars.var \<times> exp) list"   and cccexp1 :: "exp \<Rightarrow> Arity \<rightarrow> (Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> CoCalls"   and cccexp2 :: "exp \<Rightarrow> Arity \<rightarrow> (Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> CoCalls"   and heap1 :: "(Vars.var \<times> exp) list" assumes "\<And>e. e \<in> snd ` set heap2 \<Longrightarrow> cccexp1 e = cccexp2 e"   and "heap1 = heap2" shows "CoCallArityAnalysis.cccFix cccexp1 heap1 = CoCallArityAnalysis.cccFix cccexp2 heap2"
rtrancl_imp_UN_relpow: fixes p :: "'b \<times> 'b"   and R :: "('b \<times> 'b) set" assumes "p \<in> R\<^sup>*" shows "p \<in> \<Union> (range ((^^) R))"
valid_W_invD_2: fixes r :: "'f"   and fl :: "bool"   and s :: "'g process_name \<Rightarrow> ('d, 'g, 'e, 'f) Model.local_state"   and p :: "'g process_name" assumes "mw_Mark r fl \<in> set (sys_mem_store_buffers p s)"   and "valid_W_inv s" shows "fl = sys_fM s \<and> r \<in> ghost_honorary_grey (s p) \<and> sys_mem_lock s = Some p \<and> white r s \<and> filter is_mw_Mark (sys_mem_store_buffers p s) = [mw_Mark r fl]"