hcju/isabelleps huggingface dataset:
cinner_cblinfun_def: fixes A :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b"   and B :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "A \<bullet>\<^sub>C B = cnj (one_dim_iso (A *\<^sub>V (1::'a))) * one_dim_iso (B *\<^sub>V (1::'a))"
less_fset_def: fixes xs :: "'b fset"   and ys :: "'b fset" shows "xs |\<subset>| ys \<equiv> xs |\<subseteq>| ys \<and> xs \<noteq> ys"
iso_verts_eq_iff: fixes G :: "('e, 'f) pre_digraph"   and hom :: "('e, 'f, 'g, 'h) digraph_isomorphism"   and u :: "'e"   and v :: "'e" assumes "pre_digraph.digraph_isomorphism G hom"   and "u \<in> verts G"   and "v \<in> verts G" shows "(iso_verts hom u = iso_verts hom v) = (u = v)"
has_vector_derivative_def: fixes f :: "real \<Rightarrow> 'b"   and f' :: "'b"   and net :: "real filter" shows "(f has_vector_derivative f') net = (f has_derivative (\<lambda>x. x *\<^sub>R f')) net"
F1: shows "\<forall>v. {R. R \<in> v} = v"
vector_export_comp: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x * top \<sqinter> y) * z = x * top \<sqinter> y * z"
map_def: fixes f :: "'c" shows "A'.map f = (if F.A.arr f then a else \<chi>.C.null)"
cut_less_absorb: fixes I :: "'a set"   and t :: "'a" shows "I \<down>< t \<down>< t = I \<down>< t"
sp_append: fixes g :: "('c \<times> nat, 'd) rgraph_scheme"   and v1 :: "'c \<times> nat"   and es1 :: "('c \<times> nat) edge list"   and es2 :: "('c \<times> nat) edge list"   and v2 :: "'c \<times> nat"   and subs :: "(('c \<times> nat) \<times> 'c \<times> nat) set" shows "SubRel.subpath g v1 (es1 @ es2) v2 subs = (\<exists>v. SubRel.subpath g v1 es1 v subs \<and> SubRel.subpath g v es2 v2 subs)"
lc_mult_scalar: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "lc (p \<odot> q) = punit.lc p * lc q"
split_asm: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'b"   and f2 :: "nat \<Rightarrow> 'b"   and nat :: "nat" shows "P (case nat of 0 \<Rightarrow> f1 | Suc x \<Rightarrow> f2 x) = (\<not> (nat = 0 \<and> \<not> P f1 \<or> (\<exists>x2. nat = Suc x2 \<and> \<not> P (f2 x2))))"
nats_of_boolss_append: fixes n :: "nat"   and bss :: "bool list list"   and bss' :: "bool list list" assumes "list_all (is_alph n) bss"   and "list_all (is_alph n) bss'" shows "nats_of_boolss n (bss @ bss') = map2 (\<lambda>x y. x + 2 ^ length bss * y) (nats_of_boolss n bss) (nats_of_boolss n bss')"
game0'_eq: shows "local.ind_cca.game \<A> = map_spmf fst (game01' oracle_decrypt0' \<A>)"
inv_dghm_ObjMap_vrange: shows "\<R>\<^sub>\<circ> (inv_dghm \<FF>\<lparr>ObjMap\<rparr>) = \<AA>\<lparr>Obj\<rparr>"
simps_15: fixes b :: "char list exp"   and e\<^sub>1 :: "char list exp"   and e\<^sub>2 :: "char list exp" shows "fv (if (b) e\<^sub>1 else e\<^sub>2) = fv b \<union> fv e\<^sub>1 \<union> fv e\<^sub>2"
fromFull_inj: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and intT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool"   and \<sigma> :: "'c"   and a :: "univ"   and b :: "univ" assumes "InfModel wtFsym wtPsym arOf resOf parOf \<Phi> intT intF intP" shows "(InfModel.fromFull intT \<sigma> a = InfModel.fromFull intT \<sigma> b) = (a = b)"
Cons_eq_append_conv: fixes x :: "'d"   and xs :: "'d list"   and ys :: "'d list"   and zs :: "'d list" shows "(x # xs = ys @ zs) = (ys = [] \<and> x # xs = zs \<or> (\<exists>ys'. x # ys' = ys \<and> xs = ys' @ zs))"
observable_transition_def: fixes P :: "'state"   and \<alpha> :: "('act, 'pred) S_action"   and P' :: "'state" shows "P \<Rightarrow>{\<alpha>}\<^sub>S P' \<equiv> \<exists>Q Q'. P \<Rightarrow>\<^sub>S Q \<and> Q \<rightarrow>\<^sub>S \<langle>\<alpha>,Q'\<rangle> \<and> Q' \<Rightarrow>\<^sub>S P'"
cases: fixes a :: "('a \<times> 'a) set" assumes "asym a" obtains R :: "('a \<times> 'a) set" where "a = R" and "\<And>a b. (a, b) \<in> R \<Longrightarrow> (b, a) \<notin> R"
assms_2: shows "Diagonal_to_Smith_row_i_PQ_JNF i bezout (P', A', Q') = (P, A, Q)"
vmc_path_axioms: shows "vmc_path G P' v0 p \<sigma>"
takefill_le': fixes n :: "nat"   and m :: "nat"   and k :: "nat"   and x :: "'b"   and l :: "'b list" assumes "n = m + k" shows "takefill x m (takefill x n l) = takefill x m l"
simps: fixes i :: "nat"   and j :: "nat" shows "local.xij_f i j = (if i < j then (local.xij_f (i + 1) j - local.xij_f i (j - 1)) / local.xd j i else local.f i)"
assms_8: shows "\<Theta>,\<Gamma> \<turnstile> mk_eq t u"
osubst_preserves_ODE_vars: fixes I :: "('sf, 'sc, 'sz) interp"   and ODE :: "('sf + 'b, 'sz) ODE"   and \<sigma> :: "'b \<Rightarrow> ('sf, 'sz) trm"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec" shows "ODE_vars I (OsubstFO ODE \<sigma>) = ODE_vars (adjointFO I \<sigma> \<nu>) ODE"
distinct_37: fixes x51 :: "'c"   and x52 :: "('a, 'b, 'c) formula"   and x61 :: "('a, 'b, 'c) hp"   and x62 :: "('a, 'b, 'c) formula" shows "Exists x51 x52 \<noteq> (\<langle> x61 \<rangle> x62)"
path_append: fixes A :: "('c, 'd) ngba"   and r :: "('c \<times> 'd) list"   and p :: "'d"   and s :: "('c \<times> 'd) list" assumes "NGBA.path A r p"   and "NGBA.path A s (target r p)" shows "NGBA.path A (r @ s) p"
HFinite_FreeUltrafilterNat_iff: fixes X :: "nat \<Rightarrow> 'a" shows "(star_n X \<in> HFinite) = (\<exists>u. \<forall>\<^sub>F n in \<U>. norm (X n) < u)"
pos_bounded: shows "\<exists>K>0. \<forall>x. norm (f' x) \<le> norm x * K"
cases: fixes a1 :: "exp"   and a2 :: "exp" assumes "a1 \<longrightarrow>* a2" obtains e :: "exp" where "a1 = e" and "a2 = e"   | e1 :: "exp" and e2 :: "exp" and e3 :: "exp" where "a1 = e1" and "a2 = e3" and "e1 \<longrightarrow> e2" and "e2 \<longrightarrow>* e3"
mon_hom_induced_section_map: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and p :: "int" assumes "section_map X Y f" shows "hom_induced p X {} Y {} f \<in> mon (homology_group p X) (homology_group p Y)"
lists_eq_set: fixes A :: "'b set" shows "lists A = {xs. set xs \<subseteq> A}"
invar'_def: fixes m :: "'d\<cdot>('e, 'f) errorT" shows "invar' m = (fmapET\<cdot>ID\<cdot>m = m)"
iTILL_inext_nth: fixes a :: "nat"   and n :: "nat" assumes "a \<le> n" shows "[\<dots>n] \<rightarrow> a = a"
d: shows "\<exists>s' r. evaluate_dec True [] env s d (s', r)"
ovsmfidiesiOssie: shows "(ov \<union> s \<union> m \<union> f\<inverse> \<union> d\<inverse> \<union> e \<union> s\<inverse>) O (s \<union> s\<inverse> \<union> e) \<subseteq> ov \<union> f\<inverse> \<union> d\<inverse> \<union> s \<union> e \<union> s\<inverse> \<union> m"
valid_index_dimsE: fixes "is" :: "nat list"   and A :: "'b tensor" assumes "is \<lhd> dims A" obtains "isa__ = []" and "dims A = []"   | "is" :: "nat list" and ds :: "nat list" and i :: "nat" and d :: "nat" where "isa__ = i # is" and "dims A = d # ds" and "is \<lhd> ds" and "i < d"
negligible_standard_hyperplane_cart: fixes k :: "'a" shows "negligible {x. x $ k = 0}"
lex_pair_def: shows "Recursive.lex_pair = less_than <*lex*> less_than"
fps_unit_dvd: fixes f :: "'a fps"   and g :: "'a fps" assumes "fps_nth f 0 \<noteq> (0::'a)" shows "f dvd g"
s2is_def: fixes s :: "state" shows "s2is s = \<lparr>iinput = inpt s, ioutput = outpt s, ichosen = chosen s, iallInput = allInput s\<rparr>"
sets_le_imp_space_le: fixes A :: "'a measure"   and B :: "'a measure" assumes "sets A \<subseteq> sets B" shows "space A \<subseteq> space B"
poincare_mapsto_def: fixes P :: "'a set"   and X0 :: "('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set"   and S :: "'a set"   and CX :: "'a set"   and Y :: "('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set" shows "poincare_mapsto P X0 S CX Y = (\<forall>(x, d)\<in>X0. returns_to P x \<and> fst ` X0 \<subseteq> S \<and> return_time P differentiable at x within S \<and> (\<exists>D. (poincare_map P has_derivative blinfun_apply D) (at x within S) \<and> (poincare_map P x, D o\<^sub>L d) \<in> Y) \<and> (\<forall>t\<in>{0<..<return_time P x}. flow0 x t \<in> CX))"
leg1_iso: shows "\<exists>\<nu>. \<guillemotleft>\<nu> : local.dom \<omega> \<Rightarrow> s\<^sub>1 \<star> w'\<guillemotright> \<and> local.iso \<nu>"
nat_add_offset_le: fixes y :: "nat"   and n :: "nat"   and x :: "nat"   and m :: "nat"   and sz :: "nat" assumes "y \<le> 2 ^ n"   and "x < 2 ^ m"   and "sz = m + n" shows "x * 2 ^ n + y \<le> 2 ^ sz"
\<T>_equals_\<Z>_induct_step: 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 u :: "nat" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel"   and "items_le k (LocalLexing.\<J> \<TT> \<RR> \<SS> Lex Sel Doc k u) = LocalLexing.Gen \<NN> \<TT> \<RR> \<SS> Doc (paths_le k (LocalLexing.\<P> \<NN> \<TT> \<RR> \<SS> Lex Sel Doc k u))"   and "LocalLexing.\<T> \<TT> \<RR> \<SS> Lex Sel Doc k u = LocalLexing.\<Z> \<NN> \<TT> \<RR> \<SS> Lex Sel Doc k u" shows "LocalLexing.\<T> \<TT> \<RR> \<SS> Lex Sel Doc k (Suc u) = LocalLexing.\<Z> \<NN> \<TT> \<RR> \<SS> Lex Sel Doc k (Suc u)"
order_sym_trans: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "attractive A less_eq"   and "less_eq x y"   and "sympartp less_eq y z"   and "x \<in> A"   and "y \<in> A"   and "z \<in> A" shows "less_eq x z"
HPhase1or2ReadThen_blocksOf: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc"   and r :: "Proc" assumes "HPhase1or2ReadThen s s' p d q" shows "blocksOf s' r \<subseteq> blocksOf s r"
stateful_policy_to_network_graph_def: fixes \<T> :: "'b stateful_policy" shows "stateful_policy_to_network_graph \<T> = \<lparr>nodes = hosts \<T>, edges = all_flows \<T>\<rparr>"
simps_2: fixes x :: "Eg1.var"   and m :: "Mode"   and mds :: "Mode \<Rightarrow> Eg1.var set" shows "A.update_modes (x -=\<^sub>m m) mds = mds(m := {y \<in> mds m. y \<noteq> x})"
has_fields_decl_above: fixes P :: "'a prog"   and C :: "String.literal"   and FDTs :: "((String.literal \<times> String.literal) \<times> ty \<times> fmod) list"   and F :: "String.literal"   and D :: "String.literal"   and Tm :: "ty \<times> fmod" assumes "P \<turnstile> C has_fields FDTs"   and "((F, D), Tm) \<in> set FDTs" shows "P \<turnstile> C \<preceq>\<^sup>* D"
abs_eq: fixes x :: "'b multiset \<times> 'b multiset" shows "mset_pos (abs_zmultiset x) = Abs_multiset (case x of (Mp, Mn) \<Rightarrow> count (Mp - Mn))"
1: shows "\<exists>\<sigma>. Unifier \<sigma> (s \<cdot> \<theta>) (t \<cdot> \<theta>)"
lprefix_llength_le: fixes xs :: "'b llist"   and ys :: "'b llist" assumes "lprefix xs ys" shows "llength xs \<le> llength ys"
ba3_double_complement: fixes x :: "'a" shows "- - x = x"
multi_step_trans: fixes e1 :: "exp"   and e2 :: "exp"   and e3 :: "exp" assumes "e1 \<longrightarrow>* e2"   and "e2 \<longrightarrow>* e3" shows "e1 \<longrightarrow>* e3"
quotient_map_eq: fixes X :: "'c topology"   and X' :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "quotient_map X X' f"   and "\<And>x. x \<in> topspace X \<Longrightarrow> f x = g x" shows "quotient_map X X' g"
sample_get: fixes p :: "int pmf"   and f :: "int \<Rightarrow> ('v \<Rightarrow> int option) \<Rightarrow> 'm" shows "sample p (\<lambda>x. get (f x)) = get (\<lambda>s. sample p (\<lambda>x. f x s))"
\<R>_closed_glob_consistent: fixes \<Gamma>' :: "'Var \<Rightarrow> 'BExp set option"   and \<S>' :: "'Var set \<times> 'Var set"   and P' :: "'BExp set" shows "closed_glob_consistent (\<R> \<Gamma>' \<S>' P')"
id_aux4: fixes p :: "('a set set, 'b) Combinators list"   and a :: "('a set set, 'c) Combinators"   and s :: "('a set set, 'b) Combinators" assumes "noDenyAll p"   and "\<And>s. s \<in> set p \<Longrightarrow> disjSD_2 a s"   and "s \<in> set (insertDenies p)"   and "\<not> member DenyAll a" shows "disjSD_2 a s"
execute_serial_plan_sas_plus_is_execute_parallel_plan_sas_plus_i: fixes s :: "'a \<Rightarrow> 'b option"   and op :: "('a, 'b) sas_plus_operator" assumes "is_operator_applicable_in s op"   and "are_operator_effects_consistent op op" shows "s \<then>\<^sub>+ op = execute_parallel_operator_sas_plus s [op]"
LLL_invD_6: fixes upw :: "bool"   and i :: "nat"   and fs :: "int vec list" assumes "LLL_invariant upw i fs" shows "length fs = m"
Try_lemma: fixes G :: "prog"   and s1' :: "st"   and a :: "loc"   and tn :: "qtname"   and L :: "lenv"   and vn :: "lname" assumes "G\<turnstile>obj_ty (the (globs s1' (Inl a)))\<preceq>Class tn"   and "(Some (Xcpt (Loc a)), s1')\<Colon>\<preceq>(G, L)"   and "wf_prog G" shows "Norm (lupd(vn\<mapsto>Addr a) s1')\<Colon>\<preceq>(G, L(vn \<mapsto> Class tn))"
iso_is_arr: fixes f :: "'a arrow_of_spans_data" assumes "V.iso f" shows "V.arr f"
demod2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<cdot> d y \<le> d z \<cdot> x" shows "d (x \<cdot> y) \<le> d z"
sup_inf_distrib1: fixes inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'d"   and y :: "'d"   and z :: "'d" assumes "class.distrib_lattice inf less_eq less sup" shows "sup x (inf y z) = inf (sup x y) (sup x z)"
validCE_not_act: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and cid :: "'a"   and t :: "nat \<Rightarrow> cnf"   and tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and t' :: "nat \<Rightarrow> 'b"   and n :: "nat"   and \<gamma> :: "(nat \<Rightarrow> 'b) \<Rightarrow> nat \<Rightarrow> bool" assumes "\<nexists>i. active cid (t i)"   and "dynamic_component.eval tCMP active cid t t' n \<gamma>" shows "\<gamma> (lnth (dynamic_component.proj tCMP active cid (inf_llist t) @\<^sub>l inf_llist t')) n"
all_possible_assignments_for_set_is: fixes \<Psi> :: "('c, 'd) sas_plus_problem" assumes "\<forall>v\<in>set (\<Psi>\<^sub>\<V>\<^sub>+). range_of \<Psi> v \<noteq> None" shows "set (all_possible_assignments_for \<Psi>) = (\<Union>v\<in>set (\<Psi>\<^sub>\<V>\<^sub>+). {(v, a) |a. a \<in> \<R>\<^sub>+ \<Psi> v})"
strong_fusion_intro: fixes F :: "i \<Rightarrow> bool"   and x :: "i" assumes "(\<forall>y. F y \<longrightarrow> P y x) \<and> (\<forall>y. P y x \<longrightarrow> (\<exists>z. F z \<and> O y z))" shows "(\<sigma>x. F x) = x"
poly_map_rev_iterateoi_axioms: shows "poly_map_rev_iterateoi \<alpha> invar list_rev_it"
IH_2: fixes Y :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm" assumes "(qX, Y) \<in> qSwapped" shows "\<forall>ys y1 y. qAFresh ys y1 Y \<longrightarrow> qInit MOD (Y #[[y1 \<and> y]]_ys) = igSubst MOD ys (igVar MOD ys y1) y (qInit MOD Y)"
coeff_poly_shift: fixes n :: "nat"   and p :: "'a poly"   and i :: "nat" shows "coeff (poly_shift n p) i = coeff p (i + n)"
SUP_apply: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'b"   and A :: "'d set"   and x :: "'c" shows "(\<Squnion>(f ` A)) x = (\<Squnion>y\<in>A. f y x)"
simps_1: fixes T :: "'b trm"   and S :: "'b trm" shows "subterm T [] S = (T = S)"
compl_struct_def: fixes compl :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> 'f) list \<times> 'g" shows "compl_struct compl = (\<forall>gs bs ps sps data. sps \<noteq> [] \<longrightarrow> set sps \<subseteq> set ps \<longrightarrow> (\<forall>d. dickson_grading d \<longrightarrow> dgrad_p_set_le d (fst ` set (fst (compl gs bs (ps -- sps) sps data))) (args_to_set (gs, bs, ps))) \<and> component_of_term ` Keys (fst ` set (fst (compl gs bs (ps -- sps) sps data))) \<subseteq> component_of_term ` Keys (args_to_set (gs, bs, ps)) \<and> 0 \<notin> fst ` set (fst (compl gs bs (ps -- sps) sps data)) \<and> (\<forall>h\<in>set (fst (compl gs bs (ps -- sps) sps data)). \<forall>b\<in>set gs \<union> set bs. fst b \<noteq> 0 \<longrightarrow> \<not> lt (fst b) adds\<^sub>t lt (fst h)))"
varsAppendFormulae: fixes formula1 :: "Literal list list"   and formula2 :: "Literal list list" shows "vars (formula1 @ formula2) = vars formula1 \<union> vars formula2"
All2_Eats: shows "{} \<turnstile> All2 i (Var sm) \<alpha> IMP \<alpha>(i::=Var sn) IMP All2 i (Eats (Var sm) (Var sn)) \<alpha>"
is_acc_run_def: fixes G :: "('b, 'c) gb_graph_rec_scheme"   and r :: "nat \<Rightarrow> 'b" assumes "gb_graph G" shows "gb_graph.is_acc_run G r \<equiv> graph_defs.is_run G r \<and> gb_graph.is_acc G r"
augment_read_only_mono: fixes \<S> :: "nat \<Rightarrow> bool option"   and \<S>' :: "nat \<Rightarrow> bool option"   and W :: "nat set"   and R :: "nat set" assumes "read_only \<S> \<subseteq> read_only \<S>'" shows "read_only (\<S> \<oplus>\<^bsub>W\<^esub> R) \<subseteq> read_only (\<S>' \<oplus>\<^bsub>W\<^esub> R)"
drop_append: shows "drop i \<alpha> = [\<alpha> ! i] @ drop (Suc i) \<alpha>"
continuous_on_o_Pair: fixes T :: "'a set"   and X :: "'b set"   and h :: "'a \<times> 'b \<Rightarrow> 'c"   and t :: "'a" assumes "continuous_on (T \<times> X) h"   and "t \<in> T" shows "continuous_on X (h \<circ> Pair t)"
negate_eq1: fixes x :: "'a" assumes "x \<in> V" shows "- x = - 1 \<cdot> x"
check_eqv_def: fixes r :: "'a rexp"   and s :: "'a rexp" shows "check_eqv r s = (let as = add_atoms r (add_atoms s []) in case closure as (init r, init s) of None \<Rightarrow> False | Some ([], x) \<Rightarrow> True | Some (a # list, x) \<Rightarrow> False)"
uconnectedV_mono: fixes a :: "'a"   and b :: "'a"   and F :: "'a uprod set"   and F' :: "'a uprod set" assumes "(a, b) \<in> Restr (uconnected F) verts"   and "F \<subseteq> F'" shows "(a, b) \<in> Restr (uconnected F') verts"
A_trusts_NS2_lemma: fixes A :: "agent"   and B :: "agent"   and evs :: "event list"   and NA :: "nat"   and NB :: "nat" assumes "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> ns_public"   and "Crypt (pubK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace> \<in> parts (knows Spy evs)"   and "Says A B (Crypt (pubK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs" shows "Says B A (Crypt (pubK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>) \<in> set evs"
Times_empty: fixes A :: "'c set"   and B :: "'d set" shows "(A \<times> B = {}) = (A = {} \<or> B = {})"
sup_top_right: fixes x :: "'b upred" shows "(x \<or> true) = true"
inp3: fixes s :: "real" assumes "0 \<le> s"   and "s \<le> t" shows "mk_v I (OVar vid1) (sol 0, bb) (sol s) \<in> Contexts I pid3 UNIV"
order_on_defs_2: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" shows "partial_order_on A r \<equiv> preorder_on A r \<and> antisym r"
rel_distinct_1: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and y21 :: "'d"   and y22 :: "'d Tree"   and y23 :: "'d Tree" shows "\<not> rel_Tree R E (T y21 y22 y23)"
above_rep_def: fixes p :: "int extended \<times> int extended" shows "above_rep p = (if is_empty_rep p then empty_rep else let (l, h) = p in (l, \<infinity>))"
simps_1: fixes p :: "fm"   and q :: "fm" shows "isrlfm (And p q) = (isrlfm p \<and> isrlfm q)"
simps_4: fixes h1 :: "'b heap"   and h2 :: "'b heap"   and v :: "'b heap"   and va :: "'b heap list" shows "sum_ub (h1 # h2 # v # va) = 2 * log 2 (real (size_hp h1 + size_hp h2 + size_hps (v # va))) - 2 * log 2 (real (size_hps (v # va))) - 2 + sum_ub (v # va)"
open_fun_def: fixes U :: "('a \<Rightarrow> 'b) set" shows "open U = openin (product_topology (\<lambda>i. euclidean) UNIV) U"
surjective: fixes r :: "('b, 'c) stateful_policy_scheme" shows "r = \<lparr>hosts = hosts r, flows_fix = flows_fix r, flows_state = flows_state r, \<dots> = stateful_policy.more r\<rparr>"
degree_le: fixes d :: "nat"   and x :: "'a pdevs" assumes "\<forall>j\<ge>d. pdevs_apply x j = (0::'a)" shows "degree x \<le> d"
wb_lens_def: fixes x :: "'b \<Longrightarrow> 'c" shows "wb_lens x \<equiv> weak_lens x \<and> wb_lens_axioms x"
unit_disc_fix_preserve_poincare_distance: fixes M :: "moebius"   and u :: "complex_homo"   and v :: "complex_homo" assumes "unit_disc_fix M"   and "u \<in> unit_disc"   and "v \<in> unit_disc" shows "poincare_distance (moebius_pt M u) (moebius_pt M v) = poincare_distance u v"
items_auth_data_in: fixes x :: "rat"   and H :: "msg set"   and y :: "rat" assumes "Pri_AgrK x \<in> items H \<or> Pri_AgrK y \<in> items H" shows "items (insert (Auth_Data x y) H) = insert (Auth_Data x y) (items ({Pri_AgrK x, Pri_AgrK y} \<union> H))"
hd_set: fixes id :: "'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'c"   and a :: "'c"   and M :: "'c list" assumes "Matrix_Tensor.mult id f"   and "x \<in> set (a # M)" shows "x = a \<or> x \<in> set M"
language_def: shows "MN.language \<equiv> {xs. MN.nextl (init DFA) xs \<in> final DFA}"
wellformed_transaction_send_receive_trm_cases_1: fixes T :: "('e, 'f, 'g, 'h) prot_transaction"   and t :: "(('e, 'f, 'g) prot_fun, (('e, 'f, 'g) prot_fun, 'f prot_atom) Term.term \<times> nat) Term.term" assumes "wellformed_transaction T"   and "t \<in> trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_receive T)" shows "receive\<langle>t\<rangle> \<in> set (unlabel (transaction_receive T))"
once: shows "form_of_lit A \<^bold>\<or> disj_of_clause (removeAll A F), \<Gamma> \<Rightarrow>\<^sub>n"
sv_TS_def: shows "sv_TS = \<lparr>init = v_init, trans = sv_trans\<rparr>"
ipurge_ref_def: fixes I :: "('c \<times> 'c) set"   and D :: "'b \<Rightarrow> 'c"   and u :: "'c"   and xs :: "'b list"   and X :: "'b set" shows "ipurge_ref I D u xs X \<equiv> {x \<in> X. (u, D x) \<notin> I \<and> (\<forall>v\<in>sinks I D u xs. (v, D x) \<notin> I)}"
simplex_likeI: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b set"   and f :: "'b \<Rightarrow> nat set"   and A :: "nat set" assumes "ordering less_eq less"   and "finite P"   and "OrderingSetIso less_eq less dual_order.greater_eq dual_order.greater P f"   and "f ` P = Pow A" shows "ordering.simplex_like less_eq less P"
reads_singleton: fixes f :: "('l, 'm, 'n) prog"   and h :: "'l"   and h' :: "'l" shows "reads {preserved f} f h h'"
rem_implicit_pres_ops_lookup_op_Some_1: fixes vars :: "(char list \<times> nat option \<times> char list list) list"   and init :: "nat list"   and goal :: "(nat \<times> nat) list"   and ops :: "(char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and name :: "char list"   and n :: "char list"   and p :: "(nat \<times> nat) list"   and vp :: "((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list"   and e :: "nat" assumes "ast_problem.lookup_operator (vars, init, goal, ops) name = Some (n, p, vp, e)" shows "ast_problem.lookup_operator (rem_implicit_pres_ops (vars, init, goal, ops)) name = Some (rem_implicit_pres (n, p, vp, e))"
if_distrib_exec_gpv: fixes callee :: "'b \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'b) spmf"   and b :: "bool"   and x :: "('a, 'c, 'd) gpv"   and y :: "('a, 'c, 'd) gpv"   and s :: "'b" shows "exec_gpv callee (if b then x else y) s = (if b then exec_gpv callee x s else exec_gpv callee y s)"
vs: shows "take i vs @ vs ! i # drop (Suc i) vs = vs"
"1_1": shows "set X \<inter> (subst_domain \<delta> \<union> range_vars \<delta>) = {}"