hcju/isabelleps huggingface dataset:
emeasure_pair_measure_alt: fixes X :: "('c \<times> 'a) set"   and N :: "'c measure" assumes "X \<in> sets (N \<Otimes>\<^sub>M M1)" shows "emeasure (N \<Otimes>\<^sub>M M1) X = \<integral>\<^sup>+ x. emeasure M1 (Pair x -` X) \<partial>N"
interp_ins_distinct: fixes op_list :: "('a \<times> 'a option) list" assumes "insert_ops op_list" shows "distinct (interp_ins op_list)"
Oops_parts_spies: fixes A :: "agent"   and Timestamp :: "msg"   and B :: "msg"   and K :: "msg"   and X :: "msg"   and evs :: "event list" assumes "Says Server A (Crypt (shrK A) \<lbrace>Timestamp, B, K, X\<rbrace>) \<in> set evs" shows "K \<in> parts (knows Spy evs)"
axioms_1: fixes G :: "'b pair_pre_digraph" assumes "pair_pseudo_graph G" shows "pair_fin_digraph G"
span_eq_real2: fixes S :: "'c set"   and T :: "'d set" assumes "real_vector_space_pair_on S T" shows "module_on.span (*\<^sub>R) = span"
iterate_sel_no_map_def: fixes it :: "('b option \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'b option \<Rightarrow> 'b option) \<Rightarrow> 'b option \<Rightarrow> 'b option"   and P :: "'b \<Rightarrow> bool" shows "iterate_sel_no_map it P = iterate_sel it (\<lambda>x. if P x then Some x else None)"
typing_alt_simps_21: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and src :: "'a expr"   and k :: "call_kind"   and op :: "(any_unop + boolean_unop + numeric_unop + string_unop + collection_unop) + (super_binop + boolean_binop + numeric_binop + string_binop + collection_binop) + (string_ternop + collection_ternop) + String.literal"   and params :: "'a expr list"   and \<tau> :: "'a OCL_Types.type" shows "(\<Gamma> \<turnstile>\<^sub>E OperationCall src k op params : \<tau>) = (\<exists>\<tau>'. \<Gamma> \<turnstile>\<^sub>E src : \<tau>' \<and> (\<exists>\<pi>. \<Gamma> \<turnstile>\<^sub>L params : \<pi> \<and> op_type op k \<tau>' \<pi> \<tau>))"
periodic_simps_3: fixes x :: "'a"   and n :: "int" shows "cos (x + of_int n * ((2::'a) * of_real pi)) = cos x"
CFGExit_axioms: shows "CFGExit sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit"
wp_is_pre: fixes c :: "com"   and Q :: "state \<Rightarrow> bool" shows "\<turnstile> {wp c Q} c {Q}"
PLemmas_55: fixes port :: "'c"   and src_net :: "'a set set"   and dest_net :: "'a set set" shows "allow_from_to_port port src_net dest_net = {pa. dest_port pa = port} \<triangleleft> allow_all_from_to src_net dest_net"
wls_freshAbs_swapAbs_exchange1: fixes us :: "'varSort"   and s :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set"   and xs :: "'varSort"   and x2 :: "'var"   and x1 :: "'var" assumes "wlsAbs (us, s) A" shows "freshAbs xs x2 (A $[x1 \<and> x2]_xs) = freshAbs xs x1 A"
HSem_fresh_cong_below: fixes S :: "'b set"   and \<Gamma> :: "('b \<times> 'a) list"   and \<rho> :: "'b \<Rightarrow> 'c"   and \<rho>' :: "'b \<Rightarrow> 'c" assumes "\<rho> f|` (S \<union> fv \<Gamma> - domA \<Gamma>) \<sqsubseteq> \<rho>' f|` (S \<union> fv \<Gamma> - domA \<Gamma>)" shows "(\<lbrace> \<Gamma> \<rbrace>\<rho>) f|` S \<sqsubseteq> (\<lbrace> \<Gamma> \<rbrace>\<rho>') f|` S"
cols_length: fixes A :: "'a Matrix.mat" shows "length (cols A) = dim_col A"
l6: shows "(abscissa X - abscissa C)\<^sup>2 + (ordinate X - ordinate C)\<^sup>2 = (abscissa D - abscissa E)\<^sup>2 + (ordinate D - ordinate E)\<^sup>2"
compactI: fixes "open" :: "'b set \<Rightarrow> bool"   and s :: "'b set" assumes "class.topological_space open"   and "\<And>C. \<lbrakk>\<forall>t\<in>C. open t; s \<subseteq> \<Union> C\<rbrakk> \<Longrightarrow> \<exists>C'\<subseteq>C. finite C' \<and> s \<subseteq> \<Union> C'" shows "topological_space.compact open s"
vars0_def: shows "vars0 = vars1 \<union> vars2"
one_not_zero: shows "\<Phi> 1 \<noteq> 0"
rbt_lookup_rbtreeify: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and kvs :: "('c \<times> 'd) list" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq (list.map fst kvs)"   and "distinct (list.map fst kvs)" shows "ord.rbt_lookup less (rbtreeify kvs) = map_of kvs"
eq_iff: fixes x :: "'a list"   and y :: "'a list" shows "(x = y) = (y \<le>lex x \<and> x \<le>lex y)"
assms: shows "fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (F \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<delta>) \<subseteq> subst_domain \<sigma>"
nrcons_invmeas: fixes t :: "('c, 'd) FingerTreeStruc"   and nd :: "('c, 'd) Node" assumes "is_measured_ftree t"   and "is_measured_node nd" shows "is_measured_ftree (nrcons t nd)"
hypnat_add_self_not_less: fixes x :: "nat star"   and y :: "nat star" shows "\<not> x + y < x"
in_listspD: fixes A :: "'a \<Rightarrow> bool"   and xs :: "'a list" assumes "listsp A xs" shows "\<forall>x\<in>set xs. A x"
size1: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Rep :: "'a \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'a" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "1 < int CARD('a)"
err_def: fixes A :: "'c set" shows "err A = insert Err {OK x |x. x \<in> A}"
update: fixes f :: "'b \<Rightarrow>f 'a"   and a :: "'b"   and b :: "'a" assumes "P f" shows "P f(a $:= b)"
T_opt_def: fixes s :: "'b list"   and rs :: "'b list" shows "T_opt s rs = \<Sqinter> {T s rs as |as. length as = length rs}"
one_cvec_def: shows "1\<^sub>v = (1, 1)"
subst_apply_pair_pair: fixes \<I> :: "'c \<Rightarrow> ('a, 'b) Term.term"   and t :: "('a, 'c) Term.term"   and s :: "('a, 'c) Term.term" shows "(t, s) \<cdot>\<^sub>p \<I> = (t \<cdot> \<I>, s \<cdot> \<I>)"
Plus_1: shows "\<guillemotleft>ACI_norm_alt r\<guillemotright> = \<guillemotleft>r\<guillemotright>"
inject_2: fixes x21 :: "'a"   and x22 :: "addr_loc"   and x23 :: "'a val"   and y21 :: "'a"   and y22 :: "addr_loc"   and y23 :: "'a val" shows "(ReadMem x21 x22 x23 = ReadMem y21 y22 y23) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23)"
lim: shows "(\<lambda>n. norm (f n) / norm (f (Suc n))) \<longlonglongrightarrow> c"
surjective: fixes r :: "('b, 'd, 'e) sa_rec_scheme" shows "r = \<lparr>g_V = g_V r, g_E = g_E r, g_V0 = g_V0 r, sa_L = sa_L r, \<dots> = sa_rec.more r\<rparr>"
pairD: fixes a :: "'c"   and b :: "'d"   and p :: "'c \<times> 'd" assumes "(a, b) = p" shows "a = fst p \<and> b = snd p"
stutter_equivE: fixes \<sigma> :: "nat \<Rightarrow> 'a"   and \<tau> :: "nat \<Rightarrow> 'a" assumes "\<sigma> \<approx> \<tau>" obtains f :: "nat \<Rightarrow> nat" and g :: "nat \<Rightarrow> nat" where "stutter_sampler f \<sigma>" and "stutter_sampler g \<tau>" and "\<sigma> \<circ> f = \<tau> \<circ> g"
f2: fixes r :: "real"   and ra :: "real" shows "\<bar>r + - ra\<bar> = \<bar>ra + - r\<bar>"
of_nat_power: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and m :: "nat"   and n :: "nat" assumes "class.semiring_1 one times plus zero" shows "semiring_1.of_nat one plus zero (m ^ n) = power.power one times (semiring_1.of_nat one plus zero m) n"
Hilbert_fun_def: fixes A :: "(('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set"   and z :: "nat" shows "Hilbert_fun A z = phull.dim (hom_deg_set z A)"
subst_overrides: fixes stop :: "'a"   and eval :: "((('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) \<times> ('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) set"   and some_val :: "'c"   and some_val' :: "'c"   and \<sigma> :: "'d \<Rightarrow> 'e option"   and \<tau> :: "'d \<Rightarrow> 'e option"   and mem :: "'d \<Rightarrow> 'e" assumes "sifum_security stop eval some_val some_val'"   and "dom \<sigma> = dom \<tau>" shows "subst \<sigma> (subst \<tau> mem) = subst \<sigma> mem"
SF_alt: fixes A :: "state \<times> state \<Rightarrow> bool"   and v :: "state \<Rightarrow> 'a" shows "\<turnstile> SF(A)_v = (\<diamond>\<box>\<not> Enabled (<A>_v) \<or> \<box>\<diamond><A>_v)"
infinite_mono: fixes S :: "'a set"   and T :: "'a set" assumes "infinite S"   and "\<And>x. x \<in> S \<Longrightarrow> x \<in> T" shows "infinite T"
xs': shows "xs = takeWhile (\<lambda>x. x \<noteq> v) xs @ xs'"
synthetic_div_0: fixes c :: "'b" shows "synthetic_div 0 c = 0"
borel_measurable_bilinear: fixes h :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and f :: "'d \<Rightarrow> 'a"   and S :: "'d set"   and g :: "'d \<Rightarrow> 'b" assumes "bilinear h"   and "f \<in> borel_measurable (lebesgue_on S)"   and "g \<in> borel_measurable (lebesgue_on S)"   and "S \<in> sets lebesgue" shows "(\<lambda>x. h (f x) (g x)) \<in> borel_measurable (lebesgue_on S)"
Rdef': shows "Rsubst R \<sigma> = (SG', C')"
sup_left_isotone: fixes sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.semilattice_sup sup less_eq less"   and "less_eq x y" shows "less_eq (sup x z) (sup y z)"
induct: fixes P1 :: "Term.var \<Rightarrow> bool"   and P2 :: "expr \<Rightarrow> bool"   and P3 :: "stmt \<Rightarrow> bool"   and P4 :: "expr list \<Rightarrow> bool"   and expr :: "expr" assumes "\<And>x. P1 (LVar x)"   and "\<And>x1 x2 x3 x4 x5. P2 x4 \<Longrightarrow> P1 ({x1,x2,x3}x4..x5)"   and "\<And>x1 x2. \<lbrakk>P2 x1; P2 x2\<rbrakk> \<Longrightarrow> P1 (x1.[x2])"   and "\<And>x1 x2. \<lbrakk>P3 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (InsInitV x1 x2)"   and "\<And>x. P2 (NewC x)"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P2 (New x1[x2])"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P2 (Cast x1 x2)"   and "\<And>x1 x2. P2 x1 \<Longrightarrow> P2 (x1 InstOf x2)"   and "\<And>x. P2 (Lit x)"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P2 (UnOp x1 x2)"   and "\<And>x1 x2 x3. \<lbrakk>P2 x2; P2 x3\<rbrakk> \<Longrightarrow> P2 (BinOp x1 x2 x3)"   and "P2 Super"   and "\<And>x. P1 x \<Longrightarrow> P2 (Acc x)"   and "\<And>x1 x2. \<lbrakk>P1 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (x1:=x2)"   and "\<And>x1 x2 x3. \<lbrakk>P2 x1; P2 x2; P2 x3\<rbrakk> \<Longrightarrow> P2 (x1 ? x2 : x3)"   and "\<And>x1 x2 x3 x4 x5 x6 x7. \<lbrakk>P2 x4; P4 x7\<rbrakk> \<Longrightarrow> P2 ({x1,x2,x3}x4\<cdot>x5( {x6}x7))"   and "\<And>x1 x2. P2 (Methd x1 x2)"   and "\<And>x1 x2. P3 x2 \<Longrightarrow> P2 (Body x1 x2)"   and "\<And>x1 x2. \<lbrakk>P3 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (InsInitE x1 x2)"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P2 (Callee x1 x2)"   and "P3 Skip"   and "\<And>x. P2 x \<Longrightarrow> P3 (Expr x)"   and "\<And>x1 x2. P3 x2 \<Longrightarrow> P3 (x1\<bullet> x2)"   and "\<And>x1 x2. \<lbrakk>P3 x1; P3 x2\<rbrakk> \<Longrightarrow> P3 (x1;; x2)"   and "\<And>x1 x2 x3. \<lbrakk>P2 x1; P3 x2; P3 x3\<rbrakk> \<Longrightarrow> P3 (If(x1) x2 Else x3)"   and "\<And>x1 x2 x3. \<lbrakk>P2 x2; P3 x3\<rbrakk> \<Longrightarrow> P3 (x1\<bullet> While(x2) x3)"   and "\<And>x. P3 (Jmp x)"   and "\<And>x. P2 x \<Longrightarrow> P3 (Throw x)"   and "\<And>x1 x2 x3 x4. \<lbrakk>P3 x1; P3 x4\<rbrakk> \<Longrightarrow> P3 (Try x1 Catch(x2 x3) x4)"   and "\<And>x1 x2. \<lbrakk>P3 x1; P3 x2\<rbrakk> \<Longrightarrow> P3 (x1 Finally x2)"   and "\<And>x1 x2. P3 x2 \<Longrightarrow> P3 (FinA x1 x2)"   and "\<And>x. P3 (Init x)"   and "P4 []"   and "\<And>x1 x2. \<lbrakk>P2 x1; P4 x2\<rbrakk> \<Longrightarrow> P4 (x1 # x2)" shows "P2 expr"
length_vec_to_iarray: fixes xa :: "('a, 'b) vec" shows "IArray.length (vec_to_iarray xa) = CARD('b)"
reachable_s_finite_thm_1_a: fixes s :: "('a, bool) fmap"   and PROB :: "(('a, bool) fmap \<times> ('a, bool) fmap) set" assumes "s \<in> valid_states PROB" shows "\<forall>l\<in>reachable_s PROB s. l \<in> valid_states PROB"
SATAxiomEq: fixes gamma :: "(signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list" shows "(proofTree (tree subs gamma) \<and> terminal subs gamma) = SATAxiom (sequent gamma)"
deg_comp_cong: fixes u :: "'a"   and v :: "'a"   and to1 :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and to2 :: "'a \<Rightarrow> 'a \<Rightarrow> order" assumes "deg_pp (fst (rep_nat_term u)) = deg_pp (fst (rep_nat_term v)) \<Longrightarrow> to1 u v = to2 u v" shows "deg_comp to1 u v = deg_comp to2 u v"
hyps: shows "oalist_inv_raw (foldr update_by_pair xs ys)"
Suc: fixes k :: "nat"   and s :: "'b"   and t :: "'b" shows "alternating_list (Suc k) s t = alternating_list k s t @ [if gcd_nat.greater_eq k 2 then s else t]"
intro: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c" assumes "vm_path G P v0"   and "vmc_path_axioms G P p \<sigma>" shows "vmc_path G P v0 p \<sigma>"
rep_eq: fixes x :: "'b list" shows "fset (fset_of_list x) = set x"
Insert_no_failure: fixes es :: "('a \<times> ('a, 'b) operation) event list"   and i :: "'a"   and e :: "'a \<times> 'b \<times> bool"   and n :: "'a option"   and j :: "nat"   and s :: "('a \<times> 'b \<times> bool) list" assumes "es @ [Deliver (i, Insert e n)] prefix of j"   and "apply_operations es = Some s" shows "\<exists>ys. Ordered_List.insert s e n = Some ys"
length_fresh_rename_idns: fixes G :: "variable set"   and n :: "nat"   and B :: "Core.term"   and insts :: "((variable \<times> typ) \<times> Core.term) list" assumes "finite G" shows "length (fresh_rename_idns n B insts G) = n"
these_def: fixes A :: "'b option set" shows "Option.these A = the ` {x \<in> A. x \<noteq> None}"
fmfilter_alt_defs_3: fixes B :: "'c fset" shows "fmdrop_fset B = fmfilter (\<lambda>a. a |\<notin>| B)"
simps_2: fixes v :: "'b tree"   and va :: "'b tree list" shows "merge [] (v # va) = v # va"
preserves_dom: fixes f :: "'d" assumes "D.arr f" shows "D.dom (\<eta> f) = D.map (D.dom f)"
total: fixes T :: "'b set"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "enumerateable T less" shows "\<forall>x\<in>T. \<forall>y\<in>T. x \<noteq> y \<longrightarrow> less x y \<or> less y x"
Diff_insert_absorb: fixes x :: "'b"   and A :: "'b set" assumes "x \<notin> A" shows "Set.insert x A - {x} = A"
gcd_ratfps_def: shows "gcd = normalization_euclidean_semiring_class.gcd"
normal_invE_1: fixes G :: "('e, 'f) monoid_scheme"   and N :: "'e set" assumes "Group.group G"   and "N \<lhd> G" shows "subgroup N G"
7: shows "(monom P (q (deg R q)) (deg R q) \<otimes>\<^bsub>P\<^esub> monom P (p (deg R p)) (deg R p)) (deg R p + deg R q) \<noteq> \<zero>"
intro: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a" assumes "\<forall>v\<in>Graph.V c - {s, t}. sum f (Graph.incoming c v) \<le> sum f (Graph.outgoing c v)" shows "Flow_axioms c s t f"
le_add_same_cancel2: fixes a :: "'k multiset"   and b :: "'k multiset" shows "(a \<subseteq># b + a) = ({#} \<subseteq># b)"
mem_UnitsI: fixes u :: "'a \<Rightarrow> 'a" assumes "transformation.invertible u"   and "u \<in> G \<rightarrow>\<^sub>E G" shows "u \<in> transformation.Sym"
onode_disconnect2: fixes \<sigma>' :: "nat \<Rightarrow> 'c"   and i :: "nat"   and \<sigma> :: "nat \<Rightarrow> 'c"   and s :: "'d"   and R :: "nat set"   and i' :: "nat"   and S :: "(((nat \<Rightarrow> 'c) \<times> 'd) \<times> 'e seq_action \<times> (nat \<Rightarrow> 'c) \<times> 'd) set" assumes "\<sigma>' i = \<sigma> i" shows "((\<sigma>, NodeS i s R), disconnect(i', i), \<sigma>', NodeS i s (R - {i'})) \<in> onode_sos S"
seqr_assoc: fixes P :: "('b, 'e) urel"   and Q :: "('e, 'd) urel"   and R :: "('d, 'c) urel" shows "(P ;; Q) ;; R = P ;; Q ;; R"
coeffs_to_int_poly: fixes x :: "'a mod_ring poly" shows "coeffs (to_int_poly x) = map to_int_mod_ring (coeffs x)"
\<pi>c: fixes Q :: "'v \<Rightarrow> enat"   and \<pi> :: "'v \<Rightarrow> 'v option"   and u :: "'v"   and v :: "'v"   and adjs :: "('v \<times> nat) list"   and x :: "'v" shows "\<pi>'gen Q \<pi> u ((v, w {u, v}) # adjs) x = (if x = v then \<pi>' Q \<pi> u v else \<pi>'gen Q \<pi> u adjs x)"
F_subset: shows "F \<subseteq> {x. 0 < x \<and> x \<le> (int p - 1) div 2}"
yy: fixes xa :: "'a" assumes "count (mset (y # ys)) xa < count (mset (x # xs)) xa" shows "gt (yy xa) xa \<and> count (mset (x # xs)) (yy xa) < count (mset (y # ys)) (yy xa)"
sublist_rev_right: fixes xs :: "'b list"   and ys :: "'b list" shows "xs \<le>f rev ys = rev xs \<le>f ys"
hiding_cont: fixes A :: "'b set"   and f :: "'c \<Rightarrow> 'b process" assumes "finite A"   and "cont f" shows "cont (\<lambda>x. f x \ A)"
that: fixes x :: "'d"   and h' :: "'a" assumes "h \<turnstile> f \<rightarrow>\<^sub>r x"   and "h \<turnstile> f \<rightarrow>\<^sub>h h'"   and "h' \<turnstile> ok g x" shows "thesis"
mId_bijec: shows "bijec\<^bsub>M,M\<^esub> mId\<^bsub>M\<^esub> "
prv_suc_cong: fixes t1 :: "'trm"   and t2 :: "'trm" assumes "t1 \<in> atrm"   and "t2 \<in> atrm"   and "prv (eql t1 t2)" shows "prv (eql (suc t1) (suc t2))"
real_divr_pos_less1_lower_bound: fixes x :: "real"   and prec :: "nat" assumes "0 < x"   and "x \<le> 1" shows "1 \<le> real_divr prec 1 x"
length_I: shows "length (sorted_list_of_set (f ` {0..<n})) = n"
rel_fun_assoc: fixes f :: "'c \<times> 'd \<Rightarrow> 'e"   and g :: "'c \<times> 'd \<Rightarrow> 'e"   and h :: "'c \<times> 'd \<Rightarrow> 'e" shows "ps_prod.rel_partial_semigroup.times_rel_fun (ps_prod.rel_partial_semigroup.times_rel_fun f g) h = ps_prod.rel_partial_semigroup.times_rel_fun f (ps_prod.rel_partial_semigroup.times_rel_fun g h)"
ct_suffix_singleton: fixes cts :: "ct list" assumes "0 < length cts" shows "ct_suffix [TAny] cts"
subst_1: fixes y :: "name"   and t' :: "trm"   and x :: "name" shows "Var x[y::=t'] = (if x = y then t' else Var x)"
pr_ideal_axioms: shows "pr_ideal R \<pp> (+) (\<cdot>) \<zero> \<one>"
norm_scaleC: fixes scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "complex \<Rightarrow> '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 dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and a :: "complex"   and x :: "'b" assumes "class.complex_normed_vector scaleR scaleC plus zero minus uminus dist norm sgn uniformity open" shows "norm (scaleC a x) = cmod a * norm x"
xfer_fair_run: fixes cs :: "nat \<Rightarrow> bool list \<times> nat \<times> (nat \<Rightarrow> thread)" assumes "C.is_fair_run cs" obtains as :: "nat \<Rightarrow> nat \<times> nat \<times> (nat \<Rightarrow> thread)" where "A.is_fair_run as" and "\<forall>i. sim_rel (as i) (cs i)"
arg_space_complete: fixes state :: "('b \<aproc> \<times> 'b \<aproc> set list \<times> ((nat \<times> char list) \<times> 'b \<Rightarrow> 'b \<aproc> set) \<times> 'b + call \<times> (nat \<Rightarrow> 'b option) \<times> ((nat \<times> char list) \<times> 'b \<Rightarrow> 'b \<aproc> set) \<times> 'b) discr"   and p :: "lambda" assumes "state \<in> arg_poss p" shows "abs_R\<cdot>state \<subseteq> arg_poss p"
join_ge2: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and join :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and Top :: "'a"   and y :: "'a"   and x :: "'a" assumes "class.SL_top le join Top" shows "le y (join x y)"
converse_rtrancl_induct3: fixes ax :: "'a"   and ay :: "'b"   and az :: "'c"   and bx :: "'a"   and "by" :: "'b"   and bz :: "'c"   and r :: "(('a \<times> 'b \<times> 'c) \<times> 'a \<times> 'b \<times> 'c) set"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "((ax, ay, az), bx, by, bz) \<in> r\<^sup>*"   and "P bx by bz"   and "\<And>a aa b ab ac ba. \<lbrakk>((a, aa, b), ab, ac, ba) \<in> r; ((ab, ac, ba), bx, by, bz) \<in> r\<^sup>*; P ab ac ba\<rbrakk> \<Longrightarrow> P a aa b" shows "P ax ay az"
assms_2: shows "theta \<subseteq> ZOretrT theta"
gtrI_s: fixes s :: "'f"   and c :: "'f multiset"   and e :: "'g"   and s' :: "'h"   and c' :: "'h multiset"   and gtrs :: "(('f \<times> 'f multiset) \<times> 'g \<times> 'h \<times> 'h multiset) set" assumes "((s, c), e, s', c') \<in> gtrs" shows "(add_mset s c, e, add_mset s' c') \<in> gtr gtrs"
projection_zero: shows "projection {0::'b} = (\<lambda>_. 0::'b)"
mkId_def: fixes h :: "(Location \<times> Class \<times> (Field \<times> Val) list) list" shows "mkId h = {(l1, l2). l1 = l2 \<and> l1 \<in> Dom h}"
skelInp_def: fixes inp :: "'d \<Rightarrow> ('e, 'f, 'g, 'h, 'i) qTerm set option" shows "skelInp inp = qSkelInp (pickInp inp)"
hypnat_diff_is_0_eq: fixes m :: "nat star"   and n :: "nat star" shows "(m - n = 0) = (m \<le> n)"
ev_inv: fixes f :: "'b stream \<Rightarrow> 'c stream"   and P :: "'c stream \<Rightarrow> bool"   and s :: "'b stream" assumes "\<And>s. f (stl s) = stl (f s)" shows "ev P (f s) = ev (\<lambda>x. P (f x)) s"
GOOD: shows "qGoodAbs (qAbs xs x X) \<or> qGoodAbs (qAbs xs x X')"
PMS_sessionK_not_spied: fixes Na :: "nat"   and Nb :: "nat"   and PMS :: "nat"   and NA :: "nat"   and NB :: "nat"   and role :: "role"   and evs :: "event list" assumes "Key (sessionK ((Na, Nb, PRF (PMS, NA, NB)), role)) \<in> parts (knows Spy evs)"   and "evs \<in> tls" shows "Nonce PMS \<in> parts (knows Spy evs)"
restrict_spmf_trivial: fixes p :: "'c spmf"   and A :: "'c set" assumes "\<And>x. x \<in> set_spmf p \<Longrightarrow> x \<in> A" shows "p \<upharpoonleft> A = p"
rank_def: fixes BinomialTree :: "('c, 'd) BinomialTree" shows "rank BinomialTree = (case BinomialTree of Node x1 x2 x3 x4 \<Rightarrow> x3)"
AbstrLevels_S9_A6: fixes i :: "AbstrLevelsID" assumes "sS9 \<in> AbstrLevel i" shows "sA6 \<notin> AbstrLevel i"
nmibnd: shows "\<forall>x. \<not> MP \<and> P x \<longrightarrow> (\<exists>u\<in>U. u \<sqsubseteq> x)"
hseqE: fixes \<nu> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.hseq \<nu> \<mu>" obtains "Span.arr \<mu>" and "Span.arr \<nu>" and "Span.src \<nu> = Span.trg \<mu>"
assms_4: shows "\<langle>w|w\<rangle> = 1"
eNorm_triangle_eps: fixes N :: "'a quasinorm"   and x' :: "'a"   and x :: "'a"   and a :: "ennreal" assumes "eNorm N (x' - x) < a"   and "defect N = 1" obtains e :: "ennreal" where "0 < e" and "\<And>y. eNorm N (y - x') < e \<Longrightarrow> eNorm N (y - x) < a"
nAct_not_active_same: fixes n :: "nat"   and n' :: "enat"   and t :: "cnf llist"   and c :: "'id" assumes "enat n \<le> n'"   and "n' - 1 < llength t"   and "\<nexists>k. enat n \<le> enat k \<and> enat k < n' \<and> \<parallel>c\<parallel>\<^bsub>lnth t k\<^esub>" shows "\<langle>c #\<^bsub>n'\<^esub>t\<rangle> = \<langle>c #\<^bsub>enat n\<^esub>t\<rangle>"
scc_of_def: fixes u :: "'a" shows "scc_of u \<equiv> {v. u \<rightarrow>\<^sup>* v \<and> v \<rightarrow>\<^sup>* u}"
a1: shows "low_equal s1 s2 \<and> t1 = global_reg_mod w n rd s1 \<and> t2 = global_reg_mod w n rd s2"
int_not_not: fixes x :: "int" shows "NOT (NOT x) = x"
bitset_2n: fixes n :: "nat" shows "bitset (2 * n) = Suc ` bitset n"
less_eq_fps_def: fixes f :: "('a, 'b) fps"   and g :: "('a, 'b) fps" shows "(f \<le> g) = (f + g = g)"
std_dual_eq_dual_space: assumes "finite B" shows "local.std_dual ` S = dual_space S"
execute_return': fixes x :: "'b"   and h :: "heap" shows "execute (return x) h = Some (x, h)"
assms_1: assumes "0 \<le> x"   and "0 \<le> y" shows "P (gcd x y)"
select_convs_3: fixes arrays :: "typerep \<Rightarrow> nat \<Rightarrow> nat list"   and refs :: "typerep \<Rightarrow> nat \<Rightarrow> nat"   and lim :: "nat"   and more :: "'a" shows "lim \<lparr>arrays = arrays, refs = refs, lim = lim, \<dots> = more\<rparr> = lim"
compatible_mult: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "h (x \<otimes> y) = h x \<otimes>\<^bsub>S\<^esub> h y"
simps_19: fixes \<pi> :: "char list \<Rightarrow> com option"   and p :: "char list"   and s :: "char list \<Rightarrow> int \<Rightarrow> int" shows "small_step \<pi> (PCall p, s) = (case \<pi> p of None \<Rightarrow> None | Some c \<Rightarrow> Some (c, s))"