hcju/isabelleps huggingface dataset:
unit_factor_0: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b" assumes "class.semidom_divide_unit_factor divide plus minus zero times one unit_factor" shows "unit_factor zero = zero"
success_run: fixes f :: "'a Heap"   and h :: "heap" assumes "success f h" obtains h' :: "heap" and r :: "'a" where "run f (Some h) (Some h') r"
inject: fixes x1 :: "int"   and x2 :: "int"   and y1 :: "int"   and y2 :: "int" shows "(X x1 x2 = X y1 y2) = (x1 = y1 \<and> x2 = y2)"
v: shows "v = ord_term_lin.max (punit.lt (rep_list (monomial (1::'b) (term_of_pair (0::'a, i)))) \<oplus> lt (monomial (1::'b) (term_of_pair (0::'a, j)))) (punit.lt (rep_list (monomial (1::'b) (term_of_pair (0::'a, j)))) \<oplus> lt (monomial (1::'b) (term_of_pair (0::'a, i))))"
upd_return: fixes i :: "nat"   and x :: "'a"   and a :: "'a array" shows "Array.upd i x a \<bind> (\<lambda>_. return a) = Array.upd i x a"
Domainp_conversep: fixes R :: "'d \<Rightarrow> 'b \<Rightarrow> bool" shows "Domainp R\<inverse>\<inverse> = Rangep R"
map_sorted_distinct_set_unique: fixes f :: "'c \<Rightarrow> 'a"   and xs :: "'c list"   and ys :: "'c list" assumes "inj_on f (set xs \<union> set ys)"   and "sorted (map f xs)"   and "distinct (map f xs)"   and "sorted (map f ys)"   and "distinct (map f ys)"   and "set xs = set ys" shows "xs = ys"
simps_2: fixes x :: "'b \<times> 'a"   and xs :: "('b \<times> 'a) list"   and a :: "'b" shows "(x \<Colon> xs)\<langle>a\<rangle>\<^sub>? = (if fst x = a then \<lfloor>snd x\<rfloor> else xs\<langle>a\<rangle>\<^sub>?)"
ToCT_def: fixes f :: "'a \<times> 'a \<times> 'a" shows "C.T.ToCT f \<equiv> if C.CCC.arr f then C.tensor (fst f) (C.tensor (fst (snd f)) (snd (snd f))) else C.null"
simps: fixes x :: "nat" shows "addr_of_array (Array x) = x"
coprime_mod_right_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and a :: "'b"   and b :: "'b" assumes "class.euclidean_semiring divide plus minus zero times one modulo euclidean_size"   and "a \<noteq> zero" shows "algebraic_semidom.coprime times one a (modulo b a) = algebraic_semidom.coprime times one a b"
bpp_closedTr: fixes A :: "'e set"   and f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and n :: "nat"   and m :: "nat" assumes "assoc_bpp A f" shows "\<forall>x y. x \<in> add_set f A n \<and> y \<in> add_set f A m \<longrightarrow> x \<^sub>f+ y \<in> add_set f A (n + m + Suc 0)"
eucl_abs: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and Basis :: "'a set"   and abs :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "class.ordered_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less" shows "abs x = comm_monoid_add.sum plus zero (\<lambda>i. scaleR \<bar>inner x i\<bar> i) Basis"
chksubs_def: fixes gt :: "('s, 'v) tm \<Rightarrow> ('s, 'v) tm \<Rightarrow> bool"   and t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" shows "chksubs gt t s = (case s of Hd hd \<Rightarrow> True | App s1 s2 \<Rightarrow> gt t s1 \<and> gt t s2)"
F_det: fixes P :: "'a process"   and Q :: "'a process" shows "F (P [+] Q) = {(s, X). s = [] \<and> (s, X) \<in> F P \<inter> F Q} \<union> {(s, X). s \<noteq> [] \<and> (s, X) \<in> F P \<union> F Q} \<union> {(s, X). s = [] \<and> s \<in> D P \<union> D Q} \<union> {(s, X). s = [] \<and> tick \<notin> X \<and> [tick] \<in> T P \<union> T Q}"
arrow_of_spans_of_maps_axioms: shows "arrow_of_spans_of_maps (\<cdot>) (\<star>) \<a> \<i> src trg s\<^sub>0 s\<^sub>1 u\<^sub>0 u\<^sub>1 \<chi>.chine"
ffilter_eq_fempty_iff_1: fixes P :: "'b \<Rightarrow> bool"   and X :: "'b fset" shows "(ffilter P X = {||}) = (\<forall>x. x |\<in>| X \<longrightarrow> \<not> P x)"
pth_c_4: fixes c :: "real"   and x :: "'b"   and w :: "'b"   and d :: "real"   and y :: "'b"   and z :: "'b" shows "c *\<^sub>R x + w + (d *\<^sub>R y + z) \<equiv> d *\<^sub>R y + (c *\<^sub>R x + w + z)"
graeffe_recursion: fixes c :: "'a"   and as :: "'a list"   and m :: "nat"   and f :: "'a poly" assumes "graeffe_poly c as m = f" shows "graeffe_poly c as (Suc m) = Polynomial.smult ((- (1::'a)) ^ degree f) (poly_square_subst (f * f \<circ>\<^sub>p [:0::'a, - (1::'a):]))"
intros: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "simple_function M f"   and "emeasure M {y \<in> space M. f y \<noteq> (0::'d)} \<noteq> \<infinity>" shows "Bochner_Integration.simple_bochner_integrable M f"
quasi_isometric_def: fixes X :: "'c set"   and Y :: "'d set" shows "quasi_isometric X Y = (\<exists>lambda C f. lambda C -quasi_isometry_between X Y f)"
rt_fresh_as_addpreRT: fixes ip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and dip :: "nat"   and npre :: "nat set" assumes "ip \<in> kD rt" shows "rt \<approx>\<^bsub>dip\<^esub> the (addpreRT rt ip npre)"
a: fixes k :: "nat" shows "integrable M (\<lambda>x. f ((T ^^ k) x))"
Lemma_5: fixes jss :: "nat list list"   and n :: "nat" assumes "\<And>f h. associative f \<Longrightarrow> map (foldl1 f \<circ> map h) jss = scanl1 f (map h [0..<n + 1])"   and "\<And>js. js \<in> set jss \<Longrightarrow> js \<noteq> []" shows "jss = ups n"
Sup_llist_def: fixes A :: "'a llist set" shows "\<Squnion> A = lSup A"
obj_class_def: fixes obj :: "obj" shows "obj_class obj = (case tag obj of CInst C \<Rightarrow> C | Arr T k \<Rightarrow> Object)"
normalize_bounds_1_def: shows "normalize_bounds_1 = normalize_bounds_1_main real_alg_precision"
distinct_9: shows "cospan_shape.Null \<noteq> cospan_shape.BT"
that: fixes r :: "real"   and a :: "real" assumes "z = complex_of_real r * (complex_of_real (cos a) + \<i> * complex_of_real (sin a))"   and "0 \<le> r"   and "0 \<le> a"   and "a < 2 * pi" shows "thesis"
inj_extend_act: fixes h :: "'c \<times> 'd \<Rightarrow> 'e" assumes "Extend h" shows "inj (extend_act h)"
integrable_density: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and g :: "'b \<Rightarrow> real" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M"   and "AE x in M. 0 \<le> g x" shows "integrable (density M (\<lambda>x. ennreal (g x))) f = integrable M (\<lambda>x. g x *\<^sub>R f x)"
lborel_distr_plus: fixes c :: "'a" shows "distr lborel borel ((+) c) = lborel"
update_def: fixes f :: "'a \<Rightarrow> 'a" shows "State_Monad.update f = State_Monad.get \<bind> State_Monad.set \<circ> f"
cfun_eqI: fixes f :: "'a \<rightarrow> 'b"   and g :: "'a \<rightarrow> 'b" assumes "\<And>x. f\<cdot>x = g\<cdot>x" shows "f = g"
flowpipe0_def: fixes X0 :: "'c set"   and hl :: "real"   and hu :: "real"   and CX :: "'c set"   and X1 :: "'c set" shows "flowpipe0 X0 hl hu CX X1 = (0 \<le> hl \<and> hl \<le> hu \<and> X0 \<subseteq> Csafe \<and> CX \<subseteq> Csafe \<and> X1 \<subseteq> Csafe \<and> (\<forall>x0\<in>X0. \<forall>h\<in>{hl..hu}. h \<in> existence_ivl0 x0 \<and> flow0 x0 h \<in> X1 \<and> (\<forall>h'\<in>{0..h}. flow0 x0 h' \<in> CX)))"
invertible_image_lemma_1: fixes a :: "'a" assumes "invertible a"   and "a \<in> M" shows "\<eta> a \<cdot>' \<eta> (inverse a) = \<one>'"
closed_Int: fixes "open" :: "'a set \<Rightarrow> bool"   and S :: "'a set"   and T :: "'a set" assumes "class.topological_space open"   and "topological_space.closed open S"   and "topological_space.closed open T" shows "topological_space.closed open (S \<inter> T)"
eval_fds_add: fixes f :: "'a fds"   and s :: "'a"   and g :: "'a fds" assumes "fds_converges f s"   and "fds_converges g s" shows "eval_fds (f + g) s = eval_fds f s + eval_fds g s"
normalize_trick: fixes p :: "'a list" shows "p = replicate (length p - length (local.normalize p)) \<zero> @ local.normalize p"
cyc_free_obtains_valuation: fixes n :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'b DBMEntry"   and v :: "'c \<Rightarrow> nat" assumes "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> \<one> \<le> len M i i xs"   and "\<forall>c. v c \<le> n \<longrightarrow> 0 < v c" shows "\<exists>u. u \<turnstile>\<^bsub>v,n\<^esub> M"
hyps: fixes i :: "nat"   and curr_p :: "'b" assumes "i < n"   and "curr_p = curr_p mod u" shows "power_polys mult_p u curr_p n ! i = curr_p * mult_p ^ i mod u"
I_def: shows "I \<equiv> \<lambda>s k. integral {0..k} (laplace_integrand f s)"
Hyp: fixes ys :: "'b list"   and tab :: "'a \<Rightarrow> 'b option" assumes "length xs = length ys" shows "dom (tab(xs [\<mapsto>] ys)) = dom tab \<union> set xs"
wellformed_tokens_def: fixes ts :: "(symbol \<times> character list) list" shows "wellformed_tokens ts = list_all wellformed_token ts"
constructive_security_sim_obsf: fixes real_resource :: "('m + 'n, 'o + 'p) resource"   and ideal_resource :: "('q + 'n, 'r + 'p) resource"   and sim :: "('m, 'o, 'q, 'r) converter"   and \<I>_real :: "('m, 'o) \<I>"   and \<I>_ideal :: "('q, 'r) \<I>"   and \<I>_common :: "('n, 'p) \<I>"   and \<A> :: "('m + 'n, 'o + 'p) distinguisher_obsf"   and adv :: "real" assumes "constructive_security_obsf real_resource ideal_resource sim \<I>_real \<I>_ideal \<I>_common \<A> adv" shows "constructive_security_sim_obsf real_resource ideal_resource sim \<I>_real \<I>_common \<A> adv"
map2_val_compat_id: shows "map2_val_compat id"
intros: fixes C :: "('a, 'b) sequent"   and A :: "('a, 'b) form"   and Ps :: "('a, 'b) sequent list"   and R :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set" assumes "C = ( \<Empt> \<Rightarrow>* \<LM> A  \<RM>)"   and "(Ps, C) \<in> R" shows "rightPrincipal (Ps, C) A R"
prules_axioms_def: fixes C_info :: "(name, dt_def) fmap"   and rs :: "(name \<times> pterm) fset" shows "prules_axioms C_info rs \<equiv> (fBall rs prule \<and> is_fmap rs) \<and> fBall rs (\<lambda>(uu_, rhs). \<not> pre_constants.shadows_consts C_info (fst |`| rs) rhs) \<and> fBall rs (\<lambda>(uu_, rhs). consts rhs |\<subseteq>| pre_constants.all_consts C_info (fst |`| rs))"
unique_connecting_path_properties_2: fixes v :: "'a"   and w :: "'a" assumes "v \<in> V"   and "w \<in> V" shows "v \<leadsto> w \<noteq> []"
store_sub2_def: fixes instr :: "sparc_operation \<times> inst_operand list"   and curr_win :: "'b word"   and rd :: "word5"   and asi :: "word8"   and address :: "word32" shows "store_sub2 instr curr_win rd asi address \<equiv> do byte_mask \<leftarrow> gets (\<lambda>s. st_byte_mask instr address); data0 \<leftarrow> gets (st_data0 instr curr_win rd address); result0 \<leftarrow> gets (memory_write asi address byte_mask data0); if result0 = None then do raise_trap data_access_exception; return () od else do new_state \<leftarrow> gets (\<lambda>s. case result0 of Some v \<Rightarrow> v); modify (\<lambda>s. new_state); if fst instr \<in> {load_store_type STD, load_store_type STDA} then do data1 \<leftarrow> gets (user_reg_val curr_win (rd OR 1)); result1 \<leftarrow> gets (memory_write asi (address + 4) 15 data1); if result1 = None then do raise_trap data_access_exception; return () od else do new_state1 \<leftarrow> gets (\<lambda>s. case result1 of Some v \<Rightarrow> v); modify (\<lambda>s. new_state1); return () od od else return () od od"
subset_empty: fixes A :: "'b set" shows "order.greater_eq {} A = (A = {})"
has_bochner_integral_divide: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "has_bochner_integral M f x" shows "has_bochner_integral M (\<lambda>x. f x / y) (x / y)"
vector_mult_closed: fixes y :: "'a"   and x :: "'a" assumes "vector y" shows "vector (x * y)"
finite_insert: fixes a :: "'a"   and A :: "'a set" shows "finite (Set.insert a A) = finite A"
obj_insert_lc: fixes f :: "Label \<Rightarrow> sterm option"   and T :: "Sigma.type"   and t :: "sterm"   and l :: "Label" assumes "lc (Obj f T)"   and "body t" shows "lc (Obj (f(l \<mapsto> t)) T)"
set_def: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<phi> :: "'a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'b"   and ba :: "'a \<times> 'a" assumes "hom_functor C S \<phi>" shows "hom_functor.set C \<phi> ba \<equiv> \<phi> (fst ba, snd ba) ` {f. partial_magma.in_hom C f (fst ba) (snd ba)}"
set_mrbst_join: fixes t' :: "'a tree"   and t1 :: "'a tree"   and t2 :: "'a tree" assumes "t' \<in> set_pmf (mrbst_join t1 t2)"   and "bst t1"   and "bst t2"   and "\<And>x y. \<lbrakk>x \<in> set_tree t1; y \<in> set_tree t2\<rbrakk> \<Longrightarrow> x < y" shows "set_tree t' = set_tree t1 \<union> set_tree t2"
div_mult2_eq: fixes c :: "'a"   and a :: "'a"   and b :: "'a" assumes "(0::'a) \<le> c" shows "a div (b * c) = a div b div c"
IH': shows "proj n B' \<in> set (tr\<^sub>p\<^sub>c (proj n S) (proj n D))"
compat2: fixes gt :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and default :: "'c"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "order_pair gt default"   and "gt x y"   and "z \<le> y" shows "gt x z"
scaleC_add_right: fixes scaleR :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and scaleC :: "complex \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "complex"   and x :: "'d"   and y :: "'d" assumes "class.complex_vector scaleR scaleC plus zero minus uminus" shows "scaleC a (plus x y) = plus (scaleC a x) (scaleC a y)"
word_log_esimps_3: fixes x :: "'b word" shows "x OR 0 = x"
u_rt: fixes y :: "'a" assumes "y \<in> ball x e \<inter> Y" shows "u y = return_time {x \<in> S. s x = 0} y"
a_Pair_inv_I: fixes \<Gamma> :: "(Syntax.var, ty) fmap"   and e\<^sub>1 :: "Syntax.term"   and e\<^sub>2 :: "Syntax.term"   and eP :: "Syntax.term"   and eV :: "Syntax.term"   and \<tau>\<^sub>1 :: "ty"   and \<tau>\<^sub>2 :: "ty" assumes "\<Gamma> \<turnstile> Syntax.Pair e\<^sub>1 e\<^sub>2, eP, eV : Syntax.Prod \<tau>\<^sub>1 \<tau>\<^sub>2" obtains eP\<^sub>1 :: "Syntax.term" and eV\<^sub>1 :: "Syntax.term" and eP\<^sub>2 :: "Syntax.term" and eV\<^sub>2 :: "Syntax.term" where "eP = Syntax.Pair eP\<^sub>1 eP\<^sub>2" and "eV = Syntax.Pair eV\<^sub>1 eV\<^sub>2" and "\<Gamma> \<turnstile> e\<^sub>1, eP\<^sub>1, eV\<^sub>1 : \<tau>\<^sub>1" and "\<Gamma> \<turnstile> e\<^sub>2, eP\<^sub>2, eV\<^sub>2 : \<tau>\<^sub>2"
less_asym: fixes x :: "('a, 'b) Term.term"   and y :: "('a, 'b) Term.term"   and P :: "bool" assumes "x \<sqsubset> y"   and "\<not> P \<Longrightarrow> y \<sqsubset> x" shows "P"
translate_plus: fixes ccw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b set"   and a :: "'b"   and x :: "'b"   and b :: "'b"   and c :: "'b" assumes "ccw_vector_space ccw S" shows "ccw (a + x) (b + x) (c + x) = ccw a b c"
M_def: fixes v :: "'a set" shows "M v = M' (LEAST l. V l = v)"
closure_scaleC: fixes a :: "complex"   and S :: "'a set" shows "closure ((*\<^sub>C) a ` S) = (*\<^sub>C) a ` closure S"
sappend_strict: shows "sappend\<cdot>\<bottom> = \<bottom>"
l_adopt_node_axioms: shows "l_adopt_node type_wf known_ptr known_ptrs get_parent adopt_node adopt_node_locs get_child_nodes get_owner_document"
sub3_divisor: fixes d :: "nat"   and l :: "nat"   and r :: "'a"   and A :: "'a mat"   and r'' :: "'a"   and A'' :: "'a mat" assumes "local.sub3 d l (r, A) = (r'', A'')"   and "l \<le> d"   and "d \<le> dim_row A"   and "l \<le> dim_col A"   and "r \<noteq> (0::'a)" shows "r'' \<noteq> (0::'a)"
map_pmf_eq_return_pmf_iff: fixes f :: "'f \<Rightarrow> 'e"   and p :: "'f pmf"   and x :: "'e" shows "(map_pmf f p = return_pmf x) = (\<forall>y\<in>set_pmf p. f y = x)"
Quotient3_rel_abs: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and Abs :: "'e \<Rightarrow> 'f"   and Rep :: "'f \<Rightarrow> 'e"   and r :: "'e"   and s :: "'e" assumes "Quotient3 R Abs Rep"   and "R r s" shows "Abs r = Abs s"
filt_equiv_measurable: fixes F :: "'b \<Rightarrow> 'c measure"   and M :: "'c measure"   and N :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and P :: "'d measure" assumes "filt_equiv F M N"   and "f \<in> M \<rightarrow>\<^sub>M P" shows "f \<in> N \<rightarrow>\<^sub>M P"
get_dom_component_def: shows "Core_DOM_DOM_Components.get_dom_component \<equiv> l_get_dom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_dom_component Core_DOM_Functions.get_root_node Core_DOM_Functions.to_tree_order"
assms_2: fixes s1 :: "'a"   and s2 :: "'b"   and x :: "'c" assumes "X s1 s2" shows "rel_spmf (\<lambda>(a, s1') (b, s2'). a = b \<and> X s1' s2') (oracle1 s1 x) (oracle2 s2 x)"
inext_ge_Max: fixes I :: "nat set"   and n :: "nat" assumes "finite I"   and "Max I \<le> n" shows "inext n I = n"
map_nth_upt_drop_take_conv: fixes N :: "nat"   and l :: "'c list"   and M :: "nat" assumes "N \<le> length l" shows "map ((!) l) [M..<N] = drop M (take N l)"
correctCompositionSecrets_def: fixes x :: "specID" shows "correctCompositionSecrets x \<equiv> subcomponents x \<noteq> {} \<longrightarrow> specSecrets x = \<Union> (specSecrets ` subcomponents x)"
auth_guard_step3: fixes s :: "l2_state"   and B :: "agent"   and A :: "agent"   and gny :: "msg"   and Ra :: "fid_t"   and Nr :: "msg" assumes "s \<in> l2_inv4"   and "s \<in> l2_inv1"   and "Insec B A \<langle>gny, hmac \<langle>Number 0, Exp Gen (NonceF (Ra $ nx)), gny, Messages.Agent B, Messages.Agent A\<rangle> (hmac (NonceF (Ra $ ni)) Nr)\<rangle> \<in> chan s"   and "guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr>"   and "can_signal s A B" shows "\<exists>Rb. guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr> \<and> in_progressS (progress s Rb) {xny, xni, xnr, xgnx, xgny, xsk} \<and> sklvl1.guessed_frame Rb xgny = Some gny \<and> sklvl1.guessed_frame Rb xnr = Some Nr \<and> sklvl1.guessed_frame Rb xni = Some (NonceF (Ra $ ni)) \<and> sklvl1.guessed_frame Rb xgnx = Some (Exp Gen (NonceF (Ra $ nx)))"
profile_swapAltsP: fixes A :: "'c set"   and Is :: "'d set"   and P :: "'d \<Rightarrow> ('c \<times> 'c) set"   and a :: "'c"   and b :: "'c" assumes "profile A Is P"   and "{a, b} \<subseteq> A" shows "profile A Is (swapAltsP P a b)"
dghm_inc_def: fixes \<BB> :: "V"   and \<CC> :: "V" shows "dghm_inc \<BB> \<CC> = [vid_on (\<BB>\<lparr>Obj\<rparr>), vid_on (\<BB>\<lparr>Arr\<rparr>), \<BB>, \<CC>]\<^sub>\<circ>"
Gamma_eq_zero_iff: fixes z :: "'b" shows "(Gamma z = (0::'b)) = (z \<in> \<int>\<^sub>\<le>\<^sub>0)"
assms_2: shows "atom y' \<sharp> (s, j, k, y)"
wgraph_specs_5: fixes \<alpha>w :: "'a \<Rightarrow> 'b set \<Rightarrow> nat"   and \<alpha>g :: "'a \<Rightarrow> 'b ugraph"   and invar :: "'a \<Rightarrow> bool"   and adj :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> nat) list"   and empty :: "'a"   and add_edge :: "'b \<times> 'b \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"   and g :: "'a"   and u :: "'b"   and v :: "'b"   and d :: "nat" assumes "adt_wgraph \<alpha>w \<alpha>g invar adj empty add_edge"   and "invar g"   and "(u, v) \<notin> edges (\<alpha>g g)"   and "u \<noteq> v" shows "invar (add_edge (u, v) d g)"
distinct_118: fixes x151 :: "floatarith"   and x152 :: "nat"   and x4 :: "floatarith" shows "x151 ^\<^sub>e x152 \<noteq> Inverse x4"
chainp_imp_relpowp: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "nat \<Rightarrow> 'b"   and j :: "nat"   and i :: "nat" assumes "\<forall>i. P (S i) (S (Suc i))" shows "(P ^^ j) (S i) (S (i + j))"
asymp_powser: shows "\<forall>n. (\<lambda>x. f x - (\<Sum>k<n. mssnth cs k * h x ^ k)) \<in> O[F](\<lambda>x. h x ^ n)"
iteratei_def: fixes list_it :: "'c \<Rightarrow> ('b list \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'b list \<Rightarrow> 'b list) \<Rightarrow> 'b list \<Rightarrow> 'b list"   and S :: "'c" shows "poly_set_iteratei_defs.iteratei list_it S \<equiv> it_to_it (list_it S)"
enat_ord_simps_6: fixes q :: "enat" shows "(\<infinity> < q) = False"
bal_simps_1: fixes xs :: "'a list" shows "bal 0 xs = (\<langle>\<rangle>, xs)"
images_are_bij: fixes g :: "'a" assumes "g \<in> carrier G" shows "bij_betw (\<phi> g) M M"
c2: shows "\<forall>p\<in>set ps. extend ( \<Phi> + \<Gamma>' \<Rightarrow>* \<Psi>1 + \<Delta>') p \<in> set Ps'"
finite_UN_I: fixes A :: "'d set"   and B :: "'d \<Rightarrow> 'e set" assumes "finite A"   and "\<And>a. a \<in> A \<Longrightarrow> finite (B a)" shows "finite (\<Union> (B ` A))"
conversep_iff: fixes r :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'c" shows "r\<inverse>\<inverse> a b = r b a"
relcomppI: fixes r :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'d"   and b :: "'e"   and s :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and c :: "'f" assumes "r a b"   and "s b c" shows "(r OO s) a c"
hindrance_by: shows "hindrance_by (\<Gamma>\<lparr>weight := weight \<Gamma> - u\<rparr>) f \<epsilon>"
monotone_ltake: shows "monotone (rel_prod (\<le>) (\<sqsubseteq>)) (\<sqsubseteq>) (\<lambda>(x, y). ltake x y)"
well_scoped: fixes v\<^sub>1 :: "'vertex"   and p\<^sub>1 :: "('form, 'var) out_port"   and v\<^sub>2 :: "'vertex"   and p\<^sub>2 :: "('form, 'var) antecedent"   and p' :: "('form, 'var) antecedent" assumes "((v\<^sub>1, p\<^sub>1), v\<^sub>2, p\<^sub>2) \<in> edges"   and "hyps (nodeOf v\<^sub>1) p\<^sub>1 = Some p'" shows "(v\<^sub>2, p\<^sub>2) = (v\<^sub>1, p') \<or> v\<^sub>2 \<in> scope (v\<^sub>1, p')"
is_acc_def: fixes m :: "'a"   and r :: "nat \<Rightarrow> 'Q" shows "gb_graph.is_acc (to_gbg_ext m) r \<equiv> \<forall>A\<in>gbg_F (to_gbg_ext m). \<exists>\<^sub>\<infinity>i. r i \<in> A"
scale_cancel_right: fixes a :: "'b"   and x :: "'b"   and b :: "'b" shows "(a * x = b * x) = (a = b \<or> x = (0::'b))"
fconverseD: fixes a :: "V"   and b :: "V"   and r :: "V" assumes "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r\<inverse>\<^sub>\<bullet>" shows "[b, a]\<^sub>\<circ> \<in>\<^sub>\<circ> r"
wtC_def: shows "Ik.wtC \<equiv> list_all Ik.wtL"
induct: fixes P :: "nat \<Rightarrow> nat \<Rightarrow> float interval list \<Rightarrow> float interval list \<Rightarrow> float poly \<Rightarrow> taylor_model \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "nat"   and a2 :: "float interval list"   and a3 :: "float interval list"   and a4 :: "float poly"   and a5 :: "taylor_model" assumes "\<And>prec ord I a c t. P prec ord I a (c)\<^sub>p t"   and "\<And>prec ord I a n t. P prec ord I a (poly.Bound n) t"   and "\<And>prec ord I a p1 p2 t. \<lbrakk>P prec ord I a p1 t; P prec ord I a p2 t\<rbrakk> \<Longrightarrow> P prec ord I a (Add p1 p2) t"   and "\<And>prec ord I a p1 p2 t. \<lbrakk>P prec ord I a p1 t; P prec ord I a p2 t\<rbrakk> \<Longrightarrow> P prec ord I a (Sub p1 p2) t"   and "\<And>prec ord I a p1 p2 t. \<lbrakk>P prec ord I a p1 t; P prec ord I a p2 t\<rbrakk> \<Longrightarrow> P prec ord I a (Mul p1 p2) t"   and "\<And>prec ord I a p t. P prec ord I a p t \<Longrightarrow> P prec ord I a (Neg p) t"   and "\<And>prec ord I a p n t. P prec ord I a p t \<Longrightarrow> P prec ord I a (Pw p n) t"   and "\<And>prec ord I a c n p t. \<lbrakk>P prec ord I a p t; \<And>x xa. \<lbrakk>x = eval_poly_at_tm prec ord I a p t; xa = tm_mul prec ord I a t x\<rbrakk> \<Longrightarrow> P prec ord I a c t\<rbrakk> \<Longrightarrow> P prec ord I a (CN c n p) t" shows "P a0 a1 a2 a3 a4 a5"
agr: fixes m :: "nat"   and n :: "nat" assumes "dec (\<sigma> m p) = Some v"   and "dec (\<sigma> n q) = Some w" shows "v = w"