hcju/isabelleps huggingface dataset:
lin_depE: fixes A :: "'a mat"   and nc :: "nat" assumes "A \<in> carrier_mat n nc"   and "lin_dep (set (cols A))"   and "distinct (cols A)" obtains v :: "'a vec" where "v \<in> carrier_vec nc" and "v \<noteq> 0\<^sub>v nc" and "A *\<^sub>v v = 0\<^sub>v n"
in_existence_between_zeroI: fixes t :: "real"   and t0 :: "real"   and x0 :: "'a"   and s :: "real" assumes "t - t0 \<in> existence_ivl0 x0"   and "s \<in> {t0--t}" shows "s - t0 \<in> existence_ivl0 x0"
simps_29: fixes x101 :: "stmt"   and x102 :: "stmt" shows "x101 Finally x102 \<noteq> Skip"
inv_f_f: fixes x :: "'a" shows "inv (inv (inv (inv (inv hom)))) (inv (inv (inv (inv hom))) x) = x"
Rh_mono: fixes h :: "'c" shows "mono (Rf (Ch h))"
classical_PP_prv: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and num :: "'b set"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and prv :: "'c \<Rightarrow> bool"   and bprv :: "'c \<Rightarrow> bool"   and enc :: "'c \<Rightarrow> 'b"   and S :: "'c"   and P :: "'c"   and \<phi> :: "'c" assumes "Goedel_Form_Classical_HBL1_rev_prv var trm fmla Var num FvarsT substT Fvars subst eql cnj imp all exi fls prv bprv enc S P"   and "\<phi> \<in> fmla"   and "Fvars \<phi> = {}"   and "prv (Syntax_with_Connectives_False.neg imp fls (Syntax_with_Connectives_False.neg imp fls (HBL1.PP var subst P (enc \<phi>))))" shows "prv (HBL1.PP var subst P (enc \<phi>))"
fds_eq_iff: fixes f :: "'b fds"   and g :: "'b fds" shows "(f = g) = (\<forall>n>0. fds_nth f n = fds_nth g n)"
diamond_imp_semi_confluence: fixes r :: "('b \<times> 'b) set" assumes "\<diamond> r" shows "r\<inverse> O r\<^sup>* \<subseteq> r\<^sup>\<down>"
rtrancl_induct: fixes a :: "'c"   and b :: "'c"   and r :: "('c \<times> 'c) set"   and P :: "'c \<Rightarrow> bool" assumes "(a, b) \<in> r\<^sup>*"   and "P a"   and "\<And>y z. \<lbrakk>(a, y) \<in> r\<^sup>*; (y, z) \<in> r; P y\<rbrakk> \<Longrightarrow> P z" shows "P b"
lem2: fixes \<Gamma> :: "'b formula set"   and F :: "'b formula" shows "\<Gamma> \<union> AX10 \<turnstile>\<^sub>H F \<^bold>\<or> \<^bold>\<not> F"
assms: shows "{$$} \<turnstile> e, eP, eV : \<tau>"
assms: fixes i :: "nat"   and j :: "nat" assumes "i < dim_row A"   and "j < dim_col A" shows "A $$ (i, j) \<in> \<int>"
that: fixes BS :: "'a tensor list" assumes "\<And>B. B \<in> set BS \<Longrightarrow> cprank_max1 B"   and "\<And>B. B \<in> set BS \<Longrightarrow> dims A = dims B"   and "listsum (dims A) BS = A"   and "length BS = n" shows "thesis"
DFS_invar_axioms_def: fixes G :: "('a, 'b) graph_rec_scheme"   and param :: "('a, ('a, 'c) state_scheme, 'c) gen_parameterization"   and s :: "('a, 'c) state_scheme" shows "DFS_invar_axioms G param s \<equiv> rwof (param_DFS_defs.init param) (param_DFS_defs.cond G param) (param_DFS_defs.step G param) s"
emb_lfinite: fixes xs :: "'b llist"   and ys :: "'b llist" assumes "emb xs ys" shows "lfinite ys = lfinite xs"
simCases: fixes Q :: "pi"   and P :: "pi"   and Rel :: "(pi \<times> pi) set" assumes "\<And>a x Q'. \<lbrakk>Q \<longmapsto> a<\<nu>x> \<prec> Q'; x \<sharp> P\<rbrakk> \<Longrightarrow> \<exists>P'. P \<Longrightarrow>a<\<nu>x> \<prec> P' \<and> (P', Q') \<in> Rel"   and "\<And>\<alpha> Q'. Q \<longmapsto> \<alpha> \<prec> Q' \<Longrightarrow> \<exists>P'. P \<Longrightarrow>\<alpha> \<prec> P' \<and> (P', Q') \<in> Rel" shows "P \<leadsto>\<guillemotleft>Rel\<guillemotright> Q"
ksf: shows "\<forall>t\<in>{0..1}. linked_paths atends g (\<lambda>x. k (t, x))"
BoxRec: fixes P :: "state \<Rightarrow> bool" shows "\<turnstile> (\<box>P) = (Init P \<and> \<box>P$)"
simps_1: fixes a :: "'b" shows "SkewBinomialHeapStruc.remove1Prio a [] = []"
ideal_eq_UNIV_iff_contains_one: fixes B :: "'b set" shows "(ideal B = UNIV) = ((1::'b) \<in> ideal B)"
dim_gen_is_basis: fixes A :: "'a vec set" assumes "finite A"   and "A \<subseteq> carrier_vec n"   and "local.span A = carrier_vec n"   and "card A \<le> local.dim" shows "basis A"
simps_2: fixes x :: "'g"   and xs :: "'g list" shows "butlast (x # xs) = (if xs = [] then [] else x # butlast xs)"
simps: fixes a1 :: "'Var \<Rightarrow> Sec option"   and a2 :: "'BExp"   and a3 :: "Sec" shows "a1 \<turnstile>\<^sub>b a2 \<in> a3  = (\<exists>\<Gamma> e. a1 = \<Gamma> \<and> a2 = e \<and> a3 = max_dom (to_total \<Gamma> ` bexp_vars e))"
permutes_image_mset: fixes p :: "'b \<Rightarrow> 'b"   and A :: "'b set" assumes "p permutes A" shows "image_mset p (mset_set A) = mset_set A"
A: fixes s :: "real" assumes "s < u"   and "a \<le> s" shows "dist (p a) (p s) \<le> d"
hcomp_obj_self: fixes a :: "('a E.term, 'a) arr" assumes "obj a" shows "a \<star> a = a"
no_redundant_set_minimal: fixes g :: "'g" assumes "\<nexists>P. redundant_set g P" shows "cytronMinimal g"
assms_3: shows "a \<in> F \<^bsub>\<o>\<^esub> A"
freshness_lemma_unique: fixes h :: "'a \<Rightarrow> 'b" assumes "\<exists>a. atom a \<sharp> (h, h a)" shows "\<exists>!x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x"
prems_1: shows "((\<mu> R\<^sub>r \<eta> \<in> old (fst n_ns) \<longrightarrow> {\<mu>, \<eta>} \<subseteq> old (fst n_ns) \<union> new (fst n_ns) \<or> \<eta> \<in> old (fst n_ns) \<union> new (fst n_ns) \<and> \<mu> R\<^sub>r \<eta> \<in> next (fst n_ns)) \<and> (\<forall>nd\<in>snd n_ns. \<mu> R\<^sub>r \<eta> \<in> old nd \<and> name nd \<in> incoming (fst n_ns) \<longrightarrow> {\<mu>, \<eta>} \<subseteq> old nd \<or> \<eta> \<in> old nd \<and> \<mu> R\<^sub>r \<eta> \<in> old (fst n_ns) \<union> new (fst n_ns))) \<and> (\<forall>nd\<in>snd n_ns. \<forall>nd'\<in>snd n_ns. \<mu> R\<^sub>r \<eta> \<in> old nd \<and> name nd \<in> incoming nd' \<longrightarrow> {\<mu>, \<eta>} \<subseteq> old nd \<or> \<eta> \<in> old nd \<and> \<mu> R\<^sub>r \<eta> \<in> old nd') \<and> ((\<forall>nm\<in>incoming (fst n_ns). nm < name (fst n_ns)) \<and> 0 < name (fst n_ns) \<and> (\<forall>q\<in>snd n_ns. name q < name (fst n_ns) \<and> (\<forall>nm\<in>incoming q. nm < name (fst n_ns)))) \<and> expand_name_ident (snd n_ns)"
antimonoD: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'e"   and x :: "'d"   and y :: "'d" assumes "class.order less_eq less"   and "order.antimono less_eq f"   and "less_eq x y" shows "f y \<le> f x"
pCons_0_0: shows "[:0::'c:] = 0"
has_domain_iff_arr: fixes f :: "'b" shows "(D.domains f \<noteq> {}) = D.arr f"
order_refl: fixes x :: "('a, 'b) Term.term" shows "x \<sqsubseteq> x"
OneAncestor_HA: fixes HA :: "('d, 'e, 'f) hierauto" shows "OneAncestor (SAs HA) (CompFun HA)"
rbt_comp_del: shows "local.rbt_comp_del = ord.rbt_del (lt_of_comp c)"
cc: fixes rs :: "'a rule list" shows "called_c' rs = called_c rs"
all_e_ubound: shows "\<forall>e>0. \<forall>\<^sub>F M in sequentially. \<forall>n. \<bar>real_of_int (B * b (n + M + 1)) / real_of_int (prod a {M..n + M + 1})\<bar> < e / 4 * 1 / 2 ^ n"
adj_mono: fixes u :: "'a"   and v :: "'a"   and H :: "('a, 'b) pre_digraph" assumes "u \<rightarrow>\<^bsub>H\<^esub> v"   and "subgraph H G" shows "u \<rightarrow> v"
comm: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry" shows "dbm_add a b = dbm_add b a"
superposition_is_sound: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and P1 :: "'b eclause"   and P2 :: "'b eclause"   and C :: "'b eclause"   and \<sigma> :: "('b \<times> 'b trm) list"   and k :: "inferences"   and C' :: "'b literal set" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms"   and "finite (cl_ecl P1)"   and "finite (cl_ecl P2)"   and "basic_superposition.superposition trm_ord sel pos_ord filter_trms P1 P2 C \<sigma> k C'" shows "set_entails_clause {cl_ecl P1, cl_ecl P2} (cl_ecl C)"
surjective: fixes r :: "('c, 'd, 'e) pre_digraph_scheme" shows "r = \<lparr>verts = verts r, arcs = arcs r, tail = tail r, head = head r, \<dots> = pre_digraph.more r\<rparr>"
simps: fixes xs :: "('b \<times> 'c) list"   and x :: "'c" shows "linear_split xs x = linear_split_help xs x []"
ApplyExplainUIPPreservedVariables: fixes state :: "State" assumes "applyExplainUIP_dom state" shows "let state' = applyExplainUIP state in getM state' = getM state \<and> getF state' = getF state \<and> getQ state' = getQ state \<and> getWatch1 state' = getWatch1 state \<and> getWatch2 state' = getWatch2 state \<and> getWatchList state' = getWatchList state \<and> getConflictFlag state' = getConflictFlag state \<and> getConflictClause state' = getConflictClause state \<and> getSATFlag state' = getSATFlag state \<and> getReason state' = getReason state"
o_def: fixes f :: "'l \<Rightarrow> 'k"   and g :: "'j \<Rightarrow> 'l" shows "f \<circ> g = (\<lambda>x. f (g x))"
variant_names_distinct: fixes S :: "variable set"   and n :: "nat"   and s :: "variable" assumes "finite S" shows "distinct (fst (variant_variables n s S))"
span_superset: fixes S :: "'a set" shows "S \<subseteq> More_Modules.ideal S"
assms: fixes x :: "nat" assumes "x \<in> free_vars e" shows "\<Gamma> x = \<Gamma>' x"
framing: fixes \<tau> :: "'d state \<times> 'd state"   and X :: "'d state \<times> 'd state \<Rightarrow> Set('e)"   and x :: "'d state \<times> 'd state \<Rightarrow> 'e"   and P :: "'d \<Rightarrow> 'e" assumes "\<tau> \<Turnstile> X->excluding\<^sub>S\<^sub>e\<^sub>t(x)->oclIsModifiedOnly()"   and "\<tau> \<Turnstile> X->forAll\<^sub>S\<^sub>e\<^sub>t(a|not (StrictRefEq\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t x a))" shows "\<tau> \<Turnstile> x@preP \<triangleq> x@postP"
sorted_domain_raw_def: fixes ko :: "'o"   and xs :: "('a \<times> 'c) list \<times> 'o" shows "sorted_domain_raw ko xs = map fst (sort_oalist_aux ko xs)"
code_lemmas_5: fixes cs :: "QDelta ns_constraint list"   and asi_tv :: "('a, ('a \<times> QDelta atom) list) mapping \<times> ((nat, QDelta) mapping \<Rightarrow> (nat, QDelta) mapping) \<times> 'a list"   and s :: "('a, QDelta) state" shows "checkpoint_simplex' (cs, asi_tv, s) = checkpoint_s s"
lookup_pair_tl': fixes xs :: "('a \<times> 'd) list"   and k :: "'a" assumes "oalist_inv_raw xs" shows "lookup_pair (tl xs) k = (if k = fst (hd xs) then 0::'d else lookup_pair xs k)"
\<PP>_are_admissible: fixes p :: "(symbol \<times> character list) list" assumes "p \<in> \<PP>" shows "admissible p"
prod_pOp_def: fixes I :: "'h set"   and A :: "'h \<Rightarrow> ('i, 'j) aGroup_scheme" shows "prod_pOp I A = (\<lambda>f\<in>carr_prodag I A. \<lambda>g\<in>carr_prodag I A. \<lambda>x\<in>I. f x \<plusminus>\<^bsub>A x\<^esub> g x)"
map_gpv'_sel: fixes f :: "'d \<Rightarrow> 'a"   and g :: "'e \<Rightarrow> 'b"   and h :: "'c \<Rightarrow> 'f"   and gpv :: "('d, 'e, 'f) gpv" shows "the_gpv (map_gpv' f g h gpv) = map_spmf (map_generat f g (map_fun h (map_gpv' f g h))) (the_gpv gpv)"
homeomorphic_rel_boundaries_same_dimension: fixes S :: "'a set"   and T :: "'b set" assumes "S homeomorphic T"   and "aff_dim S = aff_dim T" shows "S - rel_interior S homeomorphic T - rel_interior T"
simps_1: fixes c :: "color"   and k :: "'b"   and v :: "'c"   and rt :: "('b, 'c) RBT_Impl.rbt" shows "rbt_min_opt (Branch c rbt.Empty k v rt) = k"
exhaust_7: fixes x :: "7" shows "x = 1 \<or> x = 2 \<or> x = 3 \<or> x = 4 \<or> x = 5 \<or> x = 6 \<or> x = 7"
le_supI: fixes x :: "'a"   and a :: "'a"   and b :: "'a" assumes "x \<le> a"   and "x \<le> b" shows "x \<le> a \<sqinter> b"
Re_leq: fixes x :: "'a" assumes "x \<in> A" shows "Re (f x) \<le> Re (g x)"
fmranE: fixes y :: "'d"   and m :: "'e \<rightharpoonup>\<^sub>f 'd" assumes "y |\<in>| fmran m" obtains x :: "'e" where "fmlookup m x = Some y"
map_is_empty_def: fixes m :: "'e \<Rightarrow> 'f option" shows "map_is_empty m = (m = Map.empty)"
simps: fixes C :: "'b literal set" shows "vars_of_cl C = {x. \<exists>L. x \<in> vars_of_lit L \<and> L \<in> C}"
fold_deriv_FBool: fixes d0 :: "'x \<Rightarrow> 'a \<Rightarrow> ('a, 'k) aformula"   and xs :: "'x list"   and b :: "bool" shows "fold (deriv d0) xs (FBool b) = FBool b"
max_min_distrib1: fixes b :: "'a"   and c :: "'a"   and a :: "'a" shows "max (min b c) a = min (max b a) (max c a)"
fun_upd_in_state_measure: fixes \<sigma> :: "nat \<Rightarrow> val"   and A :: "nat set"   and y :: "val"   and x :: "nat" assumes "\<sigma> \<in> space (state_measure A \<Gamma>)"   and "y \<in> space (stock_measure (\<Gamma> x))" shows "\<sigma>(x := y) \<in> space (state_measure (insert x A) \<Gamma>)"
commutativities_1: shows "\<mu>.cod.leg0 \<cdot> \<mu>.chine \<cdot> \<p>\<^sub>1[Leg0 (Dom (local.dom \<mu>)), Leg1 (Dom (local.dom \<nu>))] \<cdot> \<p>\<^sub>1[Leg0 (Dom (local.dom \<nu>)) \<cdot> \<p>\<^sub>0[Leg0 (Dom (local.dom \<mu>)), Leg1 (Dom (local.dom \<nu>))], Leg1 (Dom (local.dom (\<pi> \<star> \<rho>)))] = \<nu>.cod.leg1 \<cdot> \<nu>.chine \<cdot> \<p>\<^sub>0[Leg0 (Dom (local.dom \<mu>)), Leg1 (Dom (local.dom \<nu>))] \<cdot> \<p>\<^sub>1[Leg0 (Dom (local.dom \<nu>)) \<cdot> \<p>\<^sub>0[Leg0 (Dom (local.dom \<mu>)), Leg1 (Dom (local.dom \<nu>))], Leg1 (Dom (local.dom (\<pi> \<star> \<rho>)))]"
neqE: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.linorder less_eq less"   and "\<not> x = y" obtains "less x y"   | "less y x"
quorum_def: fixes Q :: "'a set" shows "i1.proj.quorum Q \<equiv> \<forall>p\<in>Q \<inter> W. \<exists>Sl\<in>project slices I\<^sub>1 p. Sl \<subseteq> Q"
multiplicity_self: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "p \<noteq> zero"   and "\<not> dvd.dvd times p one" shows "factorial_semiring.multiplicity times one p p = 1"
insert_select2: fixes l1 :: "'g"   and l2 :: "'g"   and f :: "'g \<Rightarrow> 'h option"   and t :: "'h" assumes "l1 \<noteq> l2" shows "(f(l1 \<mapsto> t)) l2 = f l2"
Scons_mono: fixes M :: "('f, 'g) Old_Datatype.node set"   and M' :: "('f, 'g) Old_Datatype.node set"   and N :: "('f, 'g) Old_Datatype.node set"   and N' :: "('f, 'g) Old_Datatype.node set" assumes "M \<subseteq> M'"   and "N \<subseteq> N'" shows "Scons M N \<subseteq> Scons M' N'"
partitioningIps_foldr: fixes ss :: "'a wordinterval list"   and ts :: "'a wordinterval list" shows "partitioningIps ss ts = foldr partIps ss ts"
h1: shows "\<forall>x. insertion (nth_default 0 (L[var := x])) (A\<^sup>2 - B\<^sup>2 * C) = Av\<^sup>2 - Bv\<^sup>2 * Cv"
swap_Sup: fixes le_b :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Y :: "'c set"   and Z :: "('c \<Rightarrow> 'a) set" assumes "Complete_Partial_Order.chain le_b Y"   and "Complete_Partial_Order.chain (fun_ord (\<le>)) Z"   and "\<And>f. f \<in> Z \<Longrightarrow> monotone le_b (\<le>) f" shows "(\<Squnion>x\<in>Z. \<Squnion> (x ` Y)) = (\<Squnion>x\<in>Y. \<Squnion>f\<in>Z. f x)"
prems: shows " N \<longrightarrow>\<^sub>1 N'"
freshStarSimps: fixes p :: "(name \<times> name) list"   and Xs :: "name set"   and Ys :: "name set"   and x :: "name"   and C :: "'a" assumes "set p \<subseteq> Xs \<times> Ys"   and "Xs \<sharp>* x"   and "Ys \<sharp>* x" shows "x \<sharp> p \<bullet> C = x \<sharp> C"
Quotient_ratfps: shows "Quotient (eq_onp (\<lambda>x. constant_term (denominator x) \<noteq> (0::'b))) Abs_ratfps Rep_ratfps cr_ratfps"
dom_rhs: shows "\<D>\<^sub>\<circ> [\<FF>\<lparr>ObjMap\<rparr>, \<FF>\<lparr>ArrMap\<rparr>, \<FF>\<lparr>HomDom\<rparr>, \<FF>\<lparr>HomCod\<rparr>]\<^sub>\<circ> = 4\<^sub>\<nat>"
case_distrib: fixes h :: "'f \<Rightarrow> 'e"   and f :: "'g \<Rightarrow> 'h \<Rightarrow> 'f"   and prod :: "'g \<times> 'h" shows "h (case prod of (x, xa) \<Rightarrow> f x xa) = (case prod of (x1, x2) \<Rightarrow> h (f x1 x2))"
simplex_map: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b set" assumes "ChamberComplexEndomorphism X f"   and "x \<in> X" shows "f ` x \<in> X"
remove_child_heap_is_wellformed_preserved: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and child :: "(_) node_ptr"   and h' :: "(_) heap" assumes "Shadow_DOM.heap_is_wellformed h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "h \<turnstile> Shadow_DOM.remove_child ptr child \<rightarrow>\<^sub>h h'" shows "Shadow_DOM.heap_is_wellformed h'"
Lambert_W_geI: fixes y :: "real"   and x :: "real" assumes "y * exp y \<le> x" shows "y \<le> Lambert_W x"
nGt_le_conv: fixes comp :: "'b \<Rightarrow> 'b \<Rightarrow> order"   and x :: "'b"   and y :: "'b" assumes "comparator comp" shows "(comp x y \<noteq> Gt) = le_of_comp comp x y"
fixed_point_char: fixes m :: "'a set" assumes "m \<in> subgroups_of_size (p ^ a)" shows "(m \<in> conjP.fixed_points) = (\<forall>g\<in>carrier (G\<lparr>carrier := P\<rparr>). conjugation_action (p ^ a) g m = m)"
last_ConsR: fixes xs :: "'e list"   and x :: "'e" assumes "xs \<noteq> []" shows "last (x # xs) = last xs"
Bernstein_nonneg: fixes x :: "real"   and n :: "nat"   and k :: "nat" assumes "0 \<le> x"   and "x \<le> 1" shows "0 \<le> Bernstein n k x"
ring_hom_zero: fixes h :: "'e \<Rightarrow> 'f"   and R :: "('e, 'g) ring_scheme"   and S :: "('f, 'h) ring_scheme" assumes "h \<in> ring_hom R S"   and "ring R"   and "ring S" shows "h \<zero>\<^bsub>R\<^esub> = \<zero>\<^bsub>S\<^esub>"
set_integral_cong_set: fixes M :: "'a measure"   and A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and B :: "'a set" assumes "set_borel_measurable M A f"   and "set_borel_measurable M B f"   and "AE x in M. (x \<in> A) = (x \<in> B)" shows "set_lebesgue_integral M B f = set_lebesgue_integral M A f"
proper_divisor_def: fixes n :: "nat"   and m :: "nat" shows "n properdiv m \<equiv> 1 \<le> n \<and> n < m \<and> n dvd m"
tensor_vec_def: fixes v1 :: "'a vec"   and v2 :: "'a vec" shows "st.tensor_vec v1 v2 = Matrix.vec st.d (\<lambda>i. v1 $ st.encode1 i * v2 $ st.encode2 i)"
fact_0: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c" assumes "class.semiring_char_0 one times plus zero" shows "semiring_char_0.fact one plus zero 0 = one"
distinct_281: fixes x13 :: "Syntax.term"   and x16 :: "Syntax.term" shows "Roll x13 \<noteq> Unauth x16"
coboundedI1: fixes a :: "'a"   and c :: "'a"   and b :: "'a" assumes "a \<in> U"   and "c \<in> U"   and "b \<in> U"   and "a \<le>\<^sub>o\<^sub>w c" shows "local.min a b \<le>\<^sub>o\<^sub>w c"
floor_divide_real_eq_div: fixes b :: "int"   and a :: "real" assumes "0 \<le> b" shows "\<lfloor>a / real_of_int b\<rfloor> = \<lfloor>a\<rfloor> div b"
n: shows "nrows A \<le> nrows (echelon_form_of A bezout)"
has_integral_def: fixes f :: "'a \<Rightarrow> 'b"   and I :: "'b"   and s :: "'a set" shows "(f has_integral I) s = (if \<exists>a b. s = cbox a b then (sum (\<lambda>(x, k). content k *\<^sub>R f x) \<longlongrightarrow> I) (division_filter s) else \<forall>e>0. \<exists>B>0. \<forall>a b. ball (0::'a) B \<subseteq> cbox a b \<longrightarrow> (\<exists>z. (sum (\<lambda>(x, k). content k *\<^sub>R (if x \<in> s then f x else (0::'b))) \<longlongrightarrow> z) (division_filter (cbox a b)) \<and> norm (z - I) < e))"
bex_triv_one_point2: fixes A :: "'d set"   and a :: "'d" shows "(\<exists>x\<in>A. a = x) = (a \<in> A)"
fiter_fun_def: fixes f :: "'d \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'f" shows "fiter_fun f g = (\<sqinter>) f \<circ> (\<circ>) g"
pdevs_domain_project_coord_pdevs_subset: fixes sctn :: "'a sctn"   and X :: "'a pdevs" shows "pdevs_domain (project_coord_pdevs sctn X) \<subseteq> pdevs_domain X"
idinfo_dec: fixes s :: "(agent \<times> msg) set"   and s' :: "(agent \<times> msg) set"   and X :: "msg"   and K :: "key"   and n :: "nat"   and Y :: "msg" assumes "s\<^sub>0 \<Turnstile> s"   and "s' = insert (Spy, X) s \<and> (Spy, Crypt K X) \<in> s \<and> (Spy, Key (InvK K)) \<in> s"   and "\<langle>n, Y\<rangle> = X" shows "\<langle>n, Y\<rangle> \<in> spied s"
S2TTP_sender_lemma: fixes evs :: "event list"   and K :: "nat"   and S :: "agent"   and AO :: "nat"   and R :: "agent"   and hs :: "msg" assumes "evs \<in> certified_mail"   and "Key K \<notin> analz (knows Spy evs)"   and "Crypt (pubK TTP) \<lbrace>Agent S, Number AO, Key K, Agent R, hs\<rbrace> \<in> used evs" shows "\<exists>m ctxt q. hs = Hash \<lbrace>Number ctxt, Nonce q, response S R q, Crypt K (Number m)\<rbrace> \<and> Says S R \<lbrace>Agent S, Agent TTP, Crypt K (Number m), Number AO, Number ctxt, Nonce q, Crypt (pubK TTP) \<lbrace>Agent S, Number AO, Key K, Agent R, hs\<rbrace>\<rbrace> \<in> set evs"
jpTracesp_jpTraces_eq: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and jview :: "'b \<Rightarrow> 'e Trace \<Rightarrow> 'f"   and envObs :: "'b \<Rightarrow> 'e \<Rightarrow> 'g"   and jviewInit :: "'b \<Rightarrow> 'g \<Rightarrow> 'f"   and jviewIncr :: "'b \<Rightarrow> 'g \<Rightarrow> 'f \<Rightarrow> 'f"   and envInit :: "'e list"   and envAction :: "'e \<Rightarrow> 'h list"   and envTrans :: "'h \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> 'e"   and jp :: "'b \<Rightarrow> ('g, 'd, 'i) Protocol" assumes "IncrEnvironment jkbp jview envObs jviewInit jviewIncr" shows "IncrEnvironment.jpTracesp envInit envAction envTrans envObs jp = (\<lambda>x. x \<in> IncrEnvironment.jpTraces envInit envAction envTrans envObs jp)"
r_cons_def: shows "r_cons \<equiv> Cn 2 r_ifz [recf.Id 2 1, Cn 2 S [Cn 2 r_prod_encode [r_constn 1 0, recf.Id 2 0]], r_cons_else]"
prodM_sprod_mem: fixes I :: "'f set"   and M :: "'f \<Rightarrow> ('g, 'a, 'h) Module_scheme"   and a :: "'a"   and m :: "'f \<Rightarrow> 'g" assumes "\<forall>i\<in>I. R module M i"   and "a \<in> carrier R"   and "m \<in> carr_prodag I M" shows "prodM_sprod R I M a m \<in> carr_prodag I M"
order_def: fixes a :: "'a"   and p :: "'a list" shows "order a p = (SOME n. [- a, 1::'a] %^ n divides p \<and> \<not> [- a, 1::'a] %^ Suc n divides p)"
le_approx1: fixes P :: "'b process"   and Q :: "'b process" assumes "P \<sqsubseteq> Q" shows "D Q \<subseteq> D P"
28339: fixes x :: "'a"   and y :: "'a" shows "- (- - (x \<sqinter> y) \<sqinter> x) = - (x \<sqinter> y)"
simps: fixes a1 :: "s_context"   and a2 :: "inst"   and a3 :: "t_context" shows "inst_typing a1 a2 a3 = (\<exists>\<S> fs tfs gs tgs i i' n j m ts. a1 = \<S> \<and> a2 = \<lparr>types = ts, funcs = fs, tab = i, mem = j, globs = gs\<rparr> \<and> a3 = \<lparr>types_t = ts, func_t = tfs, global = tgs, table = n, memory = m, local = [], label = [], return = None\<rparr> \<and> list_all2 (funci_agree (s_funcs \<S>)) fs tfs \<and> list_all2 (globi_agree (s_globs \<S>)) gs tgs \<and> (i = Some i' \<and> i' < length (s_tab \<S>) \<and> s_tab \<S> ! i' = the n \<or> i = None \<and> n = None) \<and> memi_agree (s_mem \<S>) j m)"
ax2: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c" shows "\<turnstile> \<box>F \<longrightarrow> \<box>[\<box>F]_v"