hcju/isabelleps huggingface dataset:
monoid_def: fixes M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c" shows "Group_Theory.monoid M composition unit \<equiv> ((\<forall>a b. a \<in> M \<longrightarrow> b \<in> M \<longrightarrow> composition a b \<in> M) \<and> unit \<in> M) \<and> (\<forall>a b c. a \<in> M \<longrightarrow> b \<in> M \<longrightarrow> c \<in> M \<longrightarrow> composition (composition a b) c = composition a (composition b c)) \<and> (\<forall>a. a \<in> M \<longrightarrow> composition unit a = a) \<and> (\<forall>a. a \<in> M \<longrightarrow> composition a unit = a)"
mI_gen_in_I1: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set"   and I :: "'a set" assumes "0 < n"   and "distinct_pds K n P"   and "ideal (O\<^bsub>K P n\<^esub>) I"   and "I \<noteq> {\<zero>\<^bsub>O\<^bsub>K P n\<^esub>\<^esub>}"   and "I \<noteq> carrier (O\<^bsub>K P n\<^esub>)" shows "mIg\<^bsub> K n P I\<^esub> \<in> I"
are_operators_interfering_def: fixes op\<^sub>1 :: "('d, 'e) strips_operator_scheme"   and op\<^sub>2 :: "('d, 'f) strips_operator_scheme" shows "are_operators_interfering op\<^sub>1 op\<^sub>2 \<equiv> list_ex (\<lambda>v. list_ex ((=) v) (delete_effects_of op\<^sub>1)) (precondition_of op\<^sub>2) \<or> list_ex (\<lambda>v. list_ex ((=) v) (precondition_of op\<^sub>1)) (delete_effects_of op\<^sub>2)"
listrel1E: fixes xs :: "'g list"   and ys :: "'g list"   and r :: "('g \<times> 'g) set" assumes "(xs, ys) \<in> listrel1 r" obtains x :: "'g" and y :: "'g" and us :: "'g list" and vs :: "'g list" where "(x, y) \<in> r" and "xs = us @ x # vs" and "ys = us @ y # vs"
sorted_sort_id: fixes xs :: "'d Word.word list" assumes "signed.sorted xs" shows "signed.sort xs = xs"
stardiff_2: fixes P :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and m :: "nat"   and ps :: "char list \<Rightarrow> int option"   and n :: "nat" assumes "(P \<and>* $ m) (ps, n)" shows "m \<le> n"
rel_pmf_bindI: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'c pmf"   and q :: "'d pmf"   and P :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'e pmf"   and g :: "'d \<Rightarrow> 'f pmf" assumes "rel_pmf R p q"   and "\<And>x y. R x y \<Longrightarrow> rel_pmf P (f x) (g y)" shows "rel_pmf P (p \<bind> f) (q \<bind> g)"
A: shows "pre_well_ordered_set A (\<sqsubseteq>)"
finite_fixp: fixes \<tau> :: "'a set \<Rightarrow> 'a set" assumes "finite UNIV"   and "mono \<tau>" shows "\<exists>i. (\<tau> ^^ i) {} = (\<tau> ^^ (i + 1)) {}"
nbnd: fixes s :: "'a set"   and f :: "'a set" assumes "s \<in> simplices"   and "face f s"   and "\<not> bnd f" shows "card {s \<in> simplices. face f s} = 2"
cross3_simps_11: fixes a :: "'d"   and b :: "'d" shows "a * b = b * a"
simps_1: shows "domain Init = {xnx, xgnx, xgny, xsk, xEnd}"
borel_measurable_INF: fixes I :: "'b set"   and F :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and M :: "'c measure" assumes "countable I"   and "\<And>i. i \<in> I \<Longrightarrow> F i \<in> borel_measurable M" shows "(\<lambda>x. INF i\<in>I. F i x) \<in> borel_measurable M"
authTicket_authentic: fixes A :: "agent"   and authK :: "nat"   and Ta :: "nat"   and evs :: "event list" assumes "Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key authK, Number Ta\<rbrace> \<in> parts (knows Spy evs)"   and "evs \<in> kerbIV_gets" shows "Says Kas A (Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Number Ta, Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key authK, Number Ta\<rbrace>\<rbrace>) \<in> set evs"
is_nth_power_mult: fixes n :: "nat"   and a :: "'b"   and b :: "'b" assumes "is_nth_power n a"   and "is_nth_power n b" shows "is_nth_power n (a * b)"
seq_char: fixes g :: "'c"   and f :: "'c" shows "C'.seq g f = (C'.arr f \<and> C'.arr g \<and> C.seq g f)"
hyps_1: fixes ws :: "'a list" assumes "n = length ws"   and "\<not> distinct ws" shows "\<exists>xs ys zs y. ws = xs @ [y] @ ys @ [y] @ zs"
R: shows "<\<Gamma>1 * Rh x x'> f' x' <\<lambda>r'. \<exists>\<^sub>Ar. \<Gamma>' * R r r' * true * \<up> (RETURN r \<le> f x)>"
jordan_matrix_dim_2: fixes n_as :: "(nat \<times> 'b) list" shows "dim_col (jordan_matrix n_as) = sum_list (map fst n_as)"
nmu_def: fixes ps :: "16 word \<times> 16 word" shows "nmu ps = ({p. simple_match_port ps p} \<noteq> UNIV)"
order_antisym: fixes x :: "'c"   and y :: "'c" assumes "x \<le> y"   and "y \<le> x" shows "x = y"
compositionality: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'e"   and set :: "'e set" shows "f -` g -` set = (g \<circ> f) -` set"
ex_Y: shows "\<forall>x. x \<in># X \<longrightarrow> (\<exists>y. y \<in># Y \<and> x < y)"
in_in_arcs_conv: fixes e :: "'b"   and G :: "('c, 'b) pre_digraph"   and v :: "'c" shows "(e \<in> in_arcs G v) = (e \<in> arcs G \<and> head G e = v)"
convex_convex_hull: fixes s :: "'b set" shows "convex (convex hull s)"
urel32_def: fixes x :: "uint32"   and y :: "int" shows "urel32 x y = (y = int_of_uint32 x \<and> y < p)"
mod_size_less: fixes b :: "'a"   and a :: "'a" assumes "b \<noteq> (0::'a)" shows "euclidean_size (a mod b) < euclidean_size b"
assms_3: shows "y \<cdot> x \<le> (x + z) \<cdot> y\<^sup>\<star>"
finite_trancl_ntranl: fixes R :: "('b \<times> 'b) set" assumes "finite R" shows "R\<^sup>+ = ntrancl (card R - 1) R"
stepsD_1: fixes E :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and xs :: "'c list" assumes "Graph_Defs.steps E (x # xs)"   and "xs \<noteq> []" shows "Graph_Defs.steps E xs"
enn2ereal: fixes x :: "ennreal" shows "enn2ereal x \<in> {x. 0 \<le> x}"
monotone_onD: fixes X :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'b" assumes "monotone_on X r s f"   and "r x y"   and "x \<in> X"   and "y \<in> X" shows "s (f x) (f y)"
hyp2_abs_def: fixes v :: "(real, 2) vec" shows "hyp2_abs v = Abs_hyp2 (proj2_pt v)"
conv_complement: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" shows "(\<ominus> x)\<^sup>t = \<ominus> x\<^sup>t"
vec0I_def: fixes ze :: "'c"   and n :: "nat" shows "vec0I ze n = replicate n ze"
acq_mode_rel: fixes c\<^sub>A :: "(Eg1.var, aexp, bexp) Stmt"   and x :: "Eg1.var"   and SomeMode :: "Mode"   and c\<^sub>A_tail :: "(Eg1.var, aexp, bexp) Stmt"   and c\<^sub>C :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and c\<^sub>C_tail :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and mds\<^sub>A :: "Mode \<Rightarrow> Eg1.var set"   and mds\<^sub>C :: "Mode \<Rightarrow> var\<^sub>C set"   and mem\<^sub>A :: "Eg1.var \<Rightarrow> nat"   and mem\<^sub>C :: "var\<^sub>C \<Rightarrow> nat" assumes "c\<^sub>A = (Skip@[x +=\<^sub>m SomeMode]) ;; c\<^sub>A_tail"   and "c\<^sub>C = (Skip@[Eg2_var\<^sub>C_of_Eg1 x +=\<^sub>m SomeMode]) ;; c\<^sub>C_tail"   and "mds\<^sub>A = mds\<^sub>A_of mds\<^sub>C"   and "mem\<^sub>A = mem\<^sub>A_of mem\<^sub>C"   and "\<forall>v\<^sub>C. v\<^sub>C \<notin> range Eg2_var\<^sub>C_of_Eg1 \<longrightarrow> v\<^sub>C \<in> mds\<^sub>C AsmNoReadOrWrite"   and "\<forall>mds\<^sub>A' mem\<^sub>A' mem\<^sub>A'' mds\<^sub>C'' mem\<^sub>C' mem\<^sub>C''. mds\<^sub>A' = mds\<^sub>A_of mds\<^sub>C'' \<and> mem\<^sub>A' = mem\<^sub>A_of mem\<^sub>C' \<and> mem\<^sub>A'' = mem\<^sub>A_of mem\<^sub>C'' \<and> eval_abv\<^sub>A \<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>A''\<rangle>\<^sub>A \<langle>Stop ;; c\<^sub>A_tail, mds\<^sub>A', mem\<^sub>A'\<rangle>\<^sub>A \<and> eval_abv\<^sub>C \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C''\<rangle>\<^sub>C \<langle>Stop ;; c\<^sub>C_tail, mds\<^sub>C'', mem\<^sub>C'\<rangle>\<^sub>C \<longrightarrow> (\<langle>Stop ;; c\<^sub>A_tail, mds\<^sub>A', mem\<^sub>A'\<rangle>\<^sub>A, \<langle>Stop ;; c\<^sub>C_tail, mds\<^sub>C'', mem\<^sub>C'\<rangle>\<^sub>C) \<in> RelS_Eg1Eg2" shows "(\<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>A\<rangle>\<^sub>A, \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C\<rangle>\<^sub>C) \<in> RelS_Eg1Eg2"
"3_1": assumes "x = y"   and "transp rel"   and "\<And>x y. x \<noteq> y \<Longrightarrow> rel x y \<or> rel y x"   and "sorted_wrt rel xs"   and "sorted_wrt rel ys" shows "sorted_wrt rel (merge_wrt rel xs ys)"
B1: shows "\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel"
assms_3: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (\<Union> (trms\<^sub>s\<^sub>t ` \<S>\<^sub>0))"
fin_append_Nil: fixes z :: "nat \<Rightarrow> 'd" shows "fin_inf_append [] z = z"
bound_mono: fixes b :: "'a"   and a :: "'a" assumes "b \<le> a" shows "bound b \<le> bound a"
proj_verts_H_def: fixes G :: "('c, 'd) pre_digraph"   and rev_G :: "'d \<Rightarrow> 'd"   and H :: "('c, 'd) pre_digraph"   and rev_H :: "'d \<Rightarrow> 'd"   and u :: "'c"   and v :: "'c"   and w :: "'c"   and uv :: "'d"   and uw :: "'d"   and vw :: "'d"   and HM :: "'d pre_map"   and x :: "'c" assumes "subdiv1_contr G rev_G H rev_H u v w uv uw vw HM" shows "subdiv1_contr.proj_verts_H u w x \<equiv> if x = w then u else x"
valid_W_inv_ghg_disjoint: fixes y :: "'d"   and s :: "'e process_name \<Rightarrow> ('f, 'e, 'g, 'd) Model.local_state"   and p :: "'e process_name"   and p0 :: "'e process_name"   and p1 :: "'e process_name" assumes "white y s"   and "sys_mem_lock s = Some p"   and "valid_W_inv s"   and "p0 \<noteq> p1" shows "WL p0 (s(p := s p\<lparr>ghost_honorary_grey := {y}\<rparr>)) \<inter> WL p1 (s(p := s p\<lparr>ghost_honorary_grey := {y}\<rparr>)) = {}"
select_convs_2: fixes access :: "acc_modi"   and cfields :: "fdecl list"   and methods :: "mdecl list"   and init :: "stmt"   and more :: "'b" shows "methods \<lparr>access = access, cfields = cfields, methods = methods, init = init, \<dots> = more\<rparr> = methods"
primes_pi_def: shows "primes_pi = prime_sum_upto (\<lambda>p. 1)"
AddSA_InitConf_IFF: fixes A :: "('a, 'b, 'c) hierauto"   and Y :: "'a set"   and SA :: "('a, 'b, 'c) seqauto"   and S :: "'a"   and X :: "'a set" assumes "InitConf A = Y"   and "States SA \<inter> HAStates A = {}"   and "S \<in> HAStates A"   and "(if S \<in> Y then insert (InitState SA) Y else Y) = X" shows "InitConf (A [++] (S, SA)) = X"
thenOrdering_EQ_iff: fixes x :: "Ordering"   and y :: "Ordering" shows "(thenOrdering\<cdot>x\<cdot>y = EQ) = (x = EQ \<and> y = EQ)"
freeword_funlift_surj: fixes f :: "'b \<Rightarrow> 'a"   and S :: "'b set" shows "freeword_funlift f ` FreeGroup S = \<langle>f ` S\<rangle>"
inline1_inline_conv_inline2: fixes gpv :: "('b, 'c2, 'r2) gpv"   and s2 :: "'s2"   and s1 :: "'s1" shows "inline1 callee1 (inline callee2 gpv s2) s1 = map_spmf (map_sum (\<lambda>(x, s2, s1). ((x, s2), s1)) (\<lambda>(x, rpv'', rpv', rpv). (x, rpv'', \<lambda>r1. rpv' r1 \<bind> (\<lambda>(r2, s2). inline callee2 (rpv r2) s2)))) (local.inline2 gpv s2 s1)"
progress: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and T :: "ty"   and l :: "char list \<Rightarrow> val option" assumes "wwf_J_prog P"   and "P \<turnstile> h \<surd>"   and "P,E,h \<turnstile> e : T"   and "\<D> e \<lfloor>dom l\<rfloor>"   and "\<not> final e" shows "\<exists>e' s'. P \<turnstile> \<langle>e,(h, l)\<rangle> \<rightarrow> \<langle>e',s'\<rangle>"
has_typ1_app_elim: fixes Ts :: "typ list"   and t :: "Core.term"   and u :: "Core.term"   and R :: "typ" assumes "Ts \<turnstile>\<^sub>\<tau> t $ u : R" obtains U :: "typ" where "typ_of1 Ts u = Some U" and "typ_of1 Ts t = Some (U \<rightarrow> R)"
l_set_disconnected_nodes_get_parent_def: fixes set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_disconnected_nodes_get_parent set_disconnected_nodes_locs get_parent_locs \<equiv> \<forall>ptr h h'. \<forall>w\<in>set_disconnected_nodes_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_parent_locs. r h h')"
field_differentiable_within_cos: fixes z :: "'a"   and S :: "'a set" shows "cos field_differentiable at z within S"
ln_1_imp_less_3': fixes x :: "nat" assumes "ln (real x) = 1" shows "x < 3"
times_x_product_rule: fixes f :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier P" shows "pderiv (f \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) = f \<oplus>\<^bsub>P\<^esub> pderiv f \<otimes>\<^bsub>P\<^esub> monom P \<one> 1"
finprod_one: fixes A :: "'b set" shows "(\<Sum>i\<in>A. (0::'a)) = (0::'a)"
sorted_map_strict: fixes g :: "nat \<Rightarrow> 'b"   and n :: "nat" assumes "strict_mono_on g {0..<n}" shows "sorted (map g [0..<n])"
simps: fixes n :: "nat"   and bs :: "bit list" shows "nat_to_bv_helper n bs = (if n = 0 then bs else nat_to_bv_helper (n div 2) ((if n mod 2 = 0 then \<zero> else \<one>) # bs))"
SAStates_ChiPlus_ChiRel_OneAncestor: fixes T :: "'a"   and SA :: "('a, 'b, 'c) seqauto"   and A :: "('a, 'b, 'c) hierauto"   and S :: "'a"   and U :: "'a" assumes "T \<in> States SA"   and "SA \<in> SAs A"   and "(S, U) \<in> ChiPlus A"   and "S \<noteq> T"   and "S \<in> States SA" shows "(T, U) \<notin> ChiRel A"
gen_g_impl_rel_ext_def: fixes Rm :: "('j \<times> 'l) set"   and Rv :: "('g \<times> 'k set) set"   and Re :: "('h \<times> ('k \<times> 'k) set) set"   and Rv0 :: "('i \<times> 'k set) set" shows "\<langle>Rm, Rv, Re, Rv0\<rangle>gen_g_impl_rel_ext \<equiv> {(\<lparr>gi_V = Vi, gi_E = Ei, gi_V0 = V0i, \<dots> = mi\<rparr>, \<lparr>g_V = V, g_E = E, g_V0 = V0, \<dots> = m\<rparr>) |Vi Ei V0i mi V E V0 m. (Vi, V) \<in> Rv \<and> (Ei, E) \<in> Re \<and> (V0i, V0) \<in> Rv0 \<and> (mi, m) \<in> Rm}"
ws_prog_def: fixes G :: "prog" shows "ws_prog G = ((\<forall>(I, i)\<in>set (ifaces G). ws_idecl G I (isuperIfs i)) \<and> (\<forall>(C, c)\<in>set (classes G). ws_cdecl G C (super c)))"
fpxs_X_power_mult: fixes n :: "nat"   and m :: "rat" shows "fpxs_X_power (rat_of_nat n * m) = fpxs_X_power m ^ n"
contour_integral_primitive: fixes S :: "complex set"   and f :: "complex \<Rightarrow> complex"   and f' :: "complex \<Rightarrow> complex"   and g :: "real \<Rightarrow> complex" assumes "\<And>x. x \<in> S \<Longrightarrow> (f has_field_derivative f' x) (at x within S)"   and "valid_path g"   and "path_image g \<subseteq> S" shows "(f' has_contour_integral f (pathfinish g) - f (pathstart g)) g"
simps: fixes q :: "'d"   and l :: "'e"   and qs :: "'d list" shows "lhs q \<rightarrow> l qs = q"
sqnf_kno: shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). l \<in> {PRreq-:16..PRreq-:18} \<longrightarrow> sqnf (rt \<xi>) (dip \<xi>) = kno)"
has_vector_derivative_of_real: fixes f :: "real \<Rightarrow> real"   and D :: "real"   and F :: "real filter" assumes "(f has_real_derivative D) F" shows "((\<lambda>x. of_real (f x)) has_vector_derivative of_real D) F"
simple_function_partition: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> ennreal"   and g :: "'b \<Rightarrow> 'c"   and v :: "'c \<Rightarrow> ennreal" assumes "simple_function M f"   and "simple_function M g"   and "\<And>x y. \<lbrakk>x \<in> space M; y \<in> space M; g x = g y\<rbrakk> \<Longrightarrow> f x = f y"   and "\<And>x. x \<in> space M \<Longrightarrow> f x = v (g x)" shows "integral\<^sup>S M f = (\<Sum>y\<in>g ` space M. v y * emeasure M {x \<in> space M. g x = y})"
bind_ask: fixes f :: "'r \<Rightarrow> 'm"   and g :: "'a \<times> 'w list \<Rightarrow> 'm" shows "bind (ask f) g = ask (\<lambda>r. bind (f r) g)"
continuous_map_def: fixes S :: "'d set"   and is_open :: "'d set \<Rightarrow> bool"   and S' :: "'e set"   and is_open' :: "'e set \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'e" shows "continuous_map S is_open S' is_open' f \<equiv> (topological_space S is_open \<and> topological_space S' is_open') \<and> Set_Theory.map f S S' \<and> continuous_map_axioms S is_open is_open' f"
"case": fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'b"   and x1 :: "'d"   and x2 :: "'e" shows "(case (x1, x2) of (x, xa) \<Rightarrow> f x xa) = f x1 x2"
leg1_in_hom_2: shows "\<guillemotleft>g\<^sub>0h\<^sub>1.p\<^sub>1 : g\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> g\<^sub>0h\<^sub>1.p\<^sub>1\<guillemotright>"
dual_def: fixes po :: "'b potype" shows "dual po = \<lparr>pset = pset po, order = (order po)\<inverse>\<rparr>"
atU_add: fixes U :: "'f set"   and c :: "'f list multiset"   and ce :: "'f list multiset" assumes "atU U c" shows "atU U (c + ce) \<and> atU U (ce + c)"
vrange_vsingleton: fixes a :: "V"   and b :: "V" shows "\<R>\<^sub>\<circ> (ZFC_in_HOL.set {\<langle>a, b\<rangle>}) = ZFC_in_HOL.set {b}"
basic_cqe_conv1_3: fixes c :: "'b" shows "(\<exists>x. poly [:c:] x \<noteq> (0::'b)) = (c \<noteq> (0::'b))"
types_preserved_store_1: fixes \<S> :: "s_context"   and \<C> :: "t_context"   and k :: "i32"   and v :: "v"   and t :: "t"   and tp :: "tp option"   and a :: "nat"   and off :: "nat"   and ts :: "t list"   and ts' :: "t list" assumes "\<S>\<bullet>\<C> \<turnstile> [$C ConstInt32 k, $C v, $Store t tp a off] : ts _> ts'" shows "\<S>\<bullet>\<C> \<turnstile> [] : ts _> ts'"
phl_skip: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "class.near_dioid_one plus times one less_eq less" shows "less_eq (times x one) (times one x)"
dom_dom: fixes f :: "'b" assumes "D.arr f" shows "D.dom (D.dom f) = D.dom f"
to_nat_mono: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Rep :: "'b \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "less a b" shows "mod_type.to_nat Rep a < mod_type.to_nat Rep b"
fres_sol: fixes y :: "'a"   and x :: "'a" shows "((y \<leftarrow> x) \<cdot> x = y) = (\<exists>z. z \<cdot> x = y)"
that: fixes n :: "nat" assumes "round (rho n) p = r"   and "fg_start_round rho p (Suc r) = Suc n"   and "fg_local A p (HOs r p) (coords (Suc r) p) (rho n) (rho (Suc n))" shows "thesis"
mem_val_alt_8_unchanged: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and pa :: "word36"   and addr :: "word36"   and val :: "word8" assumes "mem_equal s1 s2 pa" shows "mem_val_alt 8 (pa AND 68719476732) (s1\<lparr>mem := (mem s1)(10 := mem s1 10(addr \<mapsto> val), 11 := (mem s1 11)(addr := None))\<rparr>) = mem_val_alt 8 (pa AND 68719476732) (s2\<lparr>mem := (mem s2)(10 := mem s2 10(addr \<mapsto> val), 11 := (mem s2 11)(addr := None))\<rparr>) \<and> mem_val_alt 8 ((pa AND 68719476732) + 1) (s1\<lparr>mem := (mem s1)(10 := mem s1 10(addr \<mapsto> val), 11 := (mem s1 11)(addr := None))\<rparr>) = mem_val_alt 8 ((pa AND 68719476732) + 1) (s2\<lparr>mem := (mem s2)(10 := mem s2 10(addr \<mapsto> val), 11 := (mem s2 11)(addr := None))\<rparr>) \<and> mem_val_alt 8 ((pa AND 68719476732) + 2) (s1\<lparr>mem := (mem s1)(10 := mem s1 10(addr \<mapsto> val), 11 := (mem s1 11)(addr := None))\<rparr>) = mem_val_alt 8 ((pa AND 68719476732) + 2) (s2\<lparr>mem := (mem s2)(10 := mem s2 10(addr \<mapsto> val), 11 := (mem s2 11)(addr := None))\<rparr>) \<and> mem_val_alt 8 ((pa AND 68719476732) + 3) (s1\<lparr>mem := (mem s1)(10 := mem s1 10(addr \<mapsto> val), 11 := (mem s1 11)(addr := None))\<rparr>) = mem_val_alt 8 ((pa AND 68719476732) + 3) (s2\<lparr>mem := (mem s2)(10 := mem s2 10(addr \<mapsto> val), 11 := (mem s2 11)(addr := None))\<rparr>)"
find_max_invar_def: fixes P :: "'a \<Rightarrow> bool"   and S :: "'a set"   and it :: "'a set"   and \<sigma> :: "'a option" shows "find_max_invar P S it \<sigma> = (case \<sigma> of None \<Rightarrow> \<forall>x\<in>S - it. \<not> P x | Some y \<Rightarrow> P y \<and> y \<in> S - it \<and> (\<forall>x\<in>S - it - {y}. \<not> P x))"
Span_in_carrier: fixes K :: "'a set"   and Us :: "'a list" assumes "K \<subseteq> carrier R"   and "set Us \<subseteq> carrier R" shows "Span K Us \<subseteq> carrier R"
map_to_set_def: fixes m :: "'d \<Rightarrow> 'e option" shows "map_to_set m = {(k, v). m k = Some v}"
new_element_get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_element_ptr :: "(_) element_ptr"   and ptr :: "(_) element_ptr"   and getter :: "('n, 'l, 'm, 'p, 'r option) RElement_scheme \<Rightarrow> 'u" assumes "h \<turnstile> new_element \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new_element \<rightarrow>\<^sub>r new_element_ptr"   and "ptr \<noteq> new_element_ptr" shows "preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter) h h'"
nonneg: fixes x :: "real" assumes "a \<le> x" shows "0 \<le> f x"
cqt_1_\<kappa>: fixes \<phi> :: "\<kappa> \<Rightarrow> \<o>"   and \<alpha> :: "\<kappa>" shows "[[(\<^bold>\<forall>\<alpha>. \<phi> (\<alpha>\<^sup>P)) \<^bold>\<rightarrow> ((\<^bold>\<exists>\<beta>. \<beta>\<^sup>P \<^bold>= \<alpha>) \<^bold>\<rightarrow> \<phi> \<alpha>)]]"
ineqi: shows "(cmod (A $$ (i, j)))\<^sup>2 \<le> (\<Sum>xa = 0..<dim_col A. (cmod (A $$ (i, xa)))\<^sup>2)"
abs_sgn: fixes k :: "'c" shows "\<bar>k\<bar> = k * sgn k"
Class_unit_normal_subgroup: shows "additive.Class \<zero> = I"
Join_SKIP_right: fixes F :: "'b program" shows "F \<squnion> \<bottom> = F"
ProcBody: fixes P :: "'d set"   and P' :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "('d set \<times> 'e \<times> 'd set \<times> 'd set) set"   and F :: "'f set"   and body :: "('d, 'e, 'f) com"   and Q :: "'d set"   and A :: "'d set"   and p :: "'e" assumes "P \<subseteq> P'"   and "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P' body Q,A"   and "\<Gamma> p = Some body" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P Call p Q,A"
that: fixes p :: "'a \<Rightarrow> 'a" assumes "p permutes A"   and "P' = (`) p ` P" shows "thesis"
pders_changes_d2: fixes r :: "real poly"   and q :: "real poly" assumes "poly r c \<noteq> 0"   and "q \<noteq> 0"   and "\<forall>x. c < x \<and> x \<le> d2 \<longrightarrow> poly r x \<noteq> 0 \<and> poly q x \<noteq> 0" shows "changes_poly_at (r # pders q) d2 = (if sign_r_pos q c = (0 < poly r c) then changes_poly_at (pders q) d2 else 1 + changes_poly_at (pders q) d2)"
induct: fixes P :: "stack_elem list \<Rightarrow> bool"   and a0 :: "stack_elem list" assumes "P []"   and "\<And>e1 e2 S. P S \<Longrightarrow> P (Alts e1 e2 # S)"   and "\<And>x S. P S \<Longrightarrow> P (Upd x # S)"   and "\<And>x S. P S \<Longrightarrow> P (Arg x # S)"   and "\<And>x S. P S \<Longrightarrow> P (Dummy x # S)" shows "P a0"
cases: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "simple_bochner_integrable M f" obtains "simple_function M f" and "emeasure M {y \<in> space M. f y \<noteq> (0::'d)} \<noteq> \<infinity>"
c0: shows "\<forall>x. insertion (nth_default 0 (xs @ x # \<Gamma>)) C = c 0"
degree_add_less: fixes a :: "'a mpoly"   and var :: "nat"   and x :: "nat"   and b :: "'a mpoly" assumes "MPoly_Type.degree a var < x"   and "MPoly_Type.degree b var < x" shows "MPoly_Type.degree (a + b) var < x"
integrable_norm_iff: fixes f :: "'d \<Rightarrow> 'e"   and M :: "'d measure" assumes "f \<in> borel_measurable M" shows "integrable M (\<lambda>x. norm (f x)) = integrable M f"
preserves_comp: fixes g :: "'c \<times> 'c"   and f :: "'c \<times> 'c" assumes "C.VV.seq g f" shows "fst (C.VV.comp g f) \<star>\<^sub>C snd (C.VV.comp g f) = (fst g \<star>\<^sub>C snd g) \<cdot>\<^sub>C (fst f \<star>\<^sub>C snd f)"
eventually_at_top_linorder: fixes P :: "'b \<Rightarrow> bool" shows "eventually P at_top = (\<exists>N. \<forall>n\<ge>N. P n)"
poly_mult_l_distr: fixes K :: "'a set"   and p1 :: "'a list"   and p2 :: "'a list"   and p3 :: "'a list" assumes "subring K R"   and "polynomial K p1"   and "polynomial K p2"   and "polynomial K p3" shows "poly_mult (poly_add p1 p2) p3 = poly_add (poly_mult p1 p3) (poly_mult p2 p3)"