hcju/isabelleps huggingface dataset:
sams_suma__lea123789: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a"   and G :: "'a"   and H :: "'a"   and I :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.SumA Bet Cong A B C D E F G H I"   and "Tarski_neutral_dimensionless.SAMS Bet Cong A B C D E F" shows "Tarski_neutral_dimensionless.LeA Bet Cong A B C G H I"
VLambda_vdoubleton: fixes a :: "V"   and b :: "V"   and f :: "V \<Rightarrow> V" shows "VLambda (set {a, b}) f = set {\<langle>a, f a\<rangle>, \<langle>b, f b\<rangle>}"
select_convs_2: fixes carrier :: "'a set"   and mult :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and more :: "'b" shows "\<one>\<^bsub>\<lparr>carrier = carrier, monoid.mult = mult, one = one, \<dots> = more\<rparr>\<^esub> = one"
a_antitone: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "ad y \<le> ad x"
Q_of_Liminf_state_inf: fixes Sts :: "('a literal multiset set \<times> 'a literal multiset set \<times> 'a literal multiset set) llist"   and Sts' :: "('a literal multiset set \<times> 'a literal multiset set \<times> 'a literal multiset set) llist" assumes "\<not> lfinite Sts"   and "emb Sts Sts'" shows "Q_of_state (Liminf_state Sts') \<subseteq> Q_of_state (Liminf_state Sts)"
exe_all_normalized_and_ex_tcsigs_imp_all_normalized_and_ex_tcsigs: fixes arss :: "(String.literal \<times> (String.literal \<times> String.literal set list) list) list"   and cs :: "(String.literal \<times> String.literal) list" assumes "exe_ars_conds arss"   and "exe_all_normalized_and_ex_tcsigs cs arss" shows "all_normalized_and_ex_tcsigs (set cs) (translate_ars arss)"
eq_iff: fixes a :: "nat"   and b :: "nat" shows "(a = b) = (gcd_nat.greater_eq b a \<and> gcd_nat.greater_eq a b)"
type_less_left_simps_4: fixes \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" shows "(Collection \<tau> < \<sigma>) = (\<exists>\<phi>. \<sigma> = OclSuper \<or> \<sigma> = Collection \<phi> \<and> \<tau> < \<phi>)"
filter_inter_mset: fixes P :: "'k \<Rightarrow> bool"   and M :: "'k multiset"   and N :: "'k multiset" shows "filter_mset P (M \<inter># N) = filter_mset P M \<inter># filter_mset P N"
sdrop_wait: fixes \<phi> :: "'b stream \<Rightarrow> bool"   and xs :: "'b stream" assumes "ev \<phi> xs" shows "\<phi> (sdrop (wait \<phi> xs) xs)"
bind_env_def: fixes bind :: "'h \<Rightarrow> ('i \<Rightarrow> 'h) \<Rightarrow> 'h" shows "bind_env bind \<equiv> rec_envT (\<lambda>x f. EnvT (\<lambda>r. bind (x r) (\<lambda>y. run_env (f y) r)))"
CnEqual1IffUIP: fixes state :: "'a State_scheme" assumes "InvariantClCharacterization (getCl state) (getC state) (getM state)"   and "InvariantClCurrentLevel (getCl state) (getM state)"   and "InvariantCnCharacterization (getCn state) (getC state) (getM state)" shows "(getCn state = 1) = isUIP (opposite (getCl state)) (getC state) (getM state)"
cases: fixes x :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) rest_scheme \<times> ('h \<times> 'a) \<times> (('i + 'c) + 'j + 'd)" obtains rest :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) rest_scheme" and state :: "'h \<times> 'a" and iadv_core :: "'i" where "x = (rest__, state__, Inl (Inl iadv_core))"   | rest :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) rest_scheme" and state :: "'h \<times> 'a" and iadv_rest :: "'c" where "x = (resta__, statea__, Inl (Inr iadv_rest))"   | rest :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) rest_scheme" and state :: "'h \<times> 'a" and iusr_core :: "'j" where "x = (restb__, stateb__, Inr (Inl iusr_core))"   | rest :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) rest_scheme" and state :: "'h \<times> 'a" and iusr_rest :: "'d" where "x = (rest, state, Inr (Inr iusr_rest))"
hom_components_def: fixes p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "hom_components p = hom_component p ` deg_pm ` keys p"
Snd: fixes X :: "msg"   and Y :: "msg"   and H :: "msg set" assumes "\<lbrace>X, Y\<rbrace> \<in> analz H" shows "Y \<in> analz H"
total_preorder_on_axioms: shows "total_preorder_on (f -` carrier) (map_relation f le)"
sexp_case_def: fixes c :: "'d \<Rightarrow> 'c"   and d :: "nat \<Rightarrow> 'c"   and e :: "('d, unit) Old_Datatype.node set \<Rightarrow> ('d, unit) Old_Datatype.node set \<Rightarrow> 'c"   and M :: "('d, unit) Old_Datatype.node set" shows "sexp_case c d e M = (THE z. (\<exists>x. M = Leaf x \<and> z = c x) \<or> (\<exists>k. M = Numb k \<and> z = d k) \<or> (\<exists>N1 N2. M = Scons N1 N2 \<and> z = e N1 N2))"
List_conv_islist_list: fixes p :: "ref"   and h :: "ref \<Rightarrow> ref"   and ps :: "ref list" shows "List p h ps = (islist p h \<and> ps = list p h)"
Hoare_unhaltI: fixes P :: "cell list \<times> cell list \<Rightarrow> bool"   and p :: "(action \<times> nat) list" assumes "\<And>l r n. P (l, r) \<Longrightarrow> \<not> is_final (steps0 (1, l, r) p n)" shows "{P} p \<up>"
assms: shows "c \<in> {f. public f} \<inter> \<C>"
mult_imp_div_pos_le: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and y :: "'a"   and x :: "'a"   and z :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero y"   and "less_eq x (times z y)" shows "less_eq (divide x y) z"
assms_4: shows "Coplanar C1 C2 A2 B2"
GreenThm_typeI_divisible_region_boundary: fixes \<gamma> :: "(int \<times> (real \<Rightarrow> real \<times> real)) set" assumes "gen_division s (cubeImage ` two_chain_typeI)"   and "two_chain_horizontal_boundary two_chain_typeI \<subseteq> \<gamma>"   and "\<gamma> \<subseteq> two_chain_boundary two_chain_typeI" shows "integral s (\<lambda>x. - partial_vector_derivative (\<lambda>a. F a \<bullet> i) j x) = one_chain_line_integral F {i} \<gamma>"
len_not_eq_0: fixes len_of :: "'c itself \<Rightarrow> nat" assumes "class.len len_of" shows "len_of TYPE('c) \<noteq> 0"
is_nat_string_def: fixes s :: "char list" shows "is_nat_string s \<equiv> \<forall>c\<in>set s. is_digit_char c"
Suc_le': 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"   and a :: "'a" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "plus a one \<noteq> zero" shows "less a (plus a one)"
hyps: shows "fact n * fact n \<le> fact (2 * n)"
disc_4: fixes x21 :: "'a"   and x22 :: "'b"   and x23 :: "'c" shows "isState (State x21 x22 x23)"
Ide_Diagonalize_Can: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "monoidal_language C"   and "monoidal_language.Can C t" shows "monoidal_language.Ide C (monoidal_language.Diagonalize C t)"
elims: fixes x :: "'a ifex"   and xa :: "'a"   and xb :: "bool"   and y :: "'a ifex" assumes "restrict_top x xa xb = y" obtains v :: "'a" and t :: "'a ifex" and e :: "'a ifex" and var :: "'a" and val :: "bool" where "x = IF v t e" and "xa = var" and "xb = val" and "y = (if v = var then if val then t else e else IF v t e)"   | uu_ :: "'a" and uv_ :: "bool" where "x = Trueif" and "xa = uu___" and "xb = uv___" and "y = Trueif"   | uu_ :: "'a" and uv_ :: "bool" where "x = Falseif" and "xa = uu_" and "xb = uv_" and "y = Falseif"
closure_of_eq: fixes X :: "'c topology"   and S :: "'c set" shows "(X closure_of S = S) = closedin X S"
left_diff_distrib_numeral: fixes a :: "'a"   and b :: "'a"   and v :: "num" shows "(a - b) * numeral v = a * numeral v - b * numeral v"
Class_is_Left_Coset: fixes K :: "'c set"   and G :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and g :: "'c" assumes "normal_subgroup K G composition unit"   and "g \<in> G" shows "equivalence.Class G (normal_subgroup.Congruence K G composition unit) g = coset_notation.Left_Coset composition g K"
append_replicate_commute: fixes n :: "nat"   and x :: "'a"   and k :: "nat" shows "x \<up> n @ x \<up> k = x \<up> k @ x \<up> n"
cases: fixes x :: "'state_t \<times> 'dom_t \<times> 'action_t list list" obtains s :: "'state_t" and d :: "'dom_t" where "x = (s__, d__, [])"   | s :: "'state_t" and d :: "'dom_t" where "x = (sa__, da__, [[]])"   | s :: "'state_t" and d :: "'dom_t" and as' :: "'action_t list" and execs' :: "'action_t list list" where "x = (sb__, db__, [] # as' # execs'__)"   | s :: "'state_t" and d :: "'dom_t" and a :: "'action_t" and as :: "'action_t list" and execs' :: "'action_t list list" where "x = (s, d, (a # as) # execs')"
UN_iff: fixes b :: "'c"   and B :: "'d \<Rightarrow> 'c set"   and A :: "'d set" shows "(b \<in> \<Union> (B ` A)) = (\<exists>x\<in>A. b \<in> B x)"
ll_on_open_it_def: fixes T :: "real set"   and f :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and X :: "'d set" shows "ll_on_open_it T f X \<equiv> ll_on_open T f X"
jnf': shows "0 \<notin> fst ` set n_as \<and> similar_mat (of_real_hom.mat_hom A) J"
insert_subset_eq_iff: fixes a :: "'b"   and A :: "'b multiset"   and B :: "'b multiset" shows "(add_mset a A \<subseteq># B) = (a \<in># B \<and> A \<subseteq># B - {#a#})"
dvd_mult_right: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a * b dvd c" shows "b dvd c"
f_def: shows "f \<equiv> \<lambda>(d1, d2). d1 * d2"
hom: fixes g :: "'g"   and g' :: "'g" assumes "g \<in> G"   and "g' \<in> G" shows "T (g + g') = T g + T g'"
wt: shows "\<forall>y.  |x\<rangle>  |x\<rangle> y \<le>  |x\<rangle> y"
fls_base_factor_fps_to_fls: fixes f :: "'a fps" shows "fls_base_factor (fps_to_fls f) = fps_to_fls (unit_factor f)"
timpl_closure'_timpls_trancl_subset: fixes c :: "('g set \<times> 'g set) set" shows "timpl_closure' (c\<^sup>+) \<subseteq> timpl_closure' c"
logical_transitivity: fixes \<Gamma> :: "(name \<times> ty) list"   and s :: "trm"   and t :: "trm"   and T :: "ty"   and u :: "trm" assumes "\<Gamma> \<turnstile> s is t : T"   and "\<Gamma> \<turnstile> t is u : T" shows "\<Gamma> \<turnstile> s is u : T"
i_expand_i_take_mod: fixes n :: "nat"   and k :: "nat"   and f :: "nat \<Rightarrow> 'b message_af" assumes "n mod k = 0" shows "f \<odot> k \<Down> n = f \<Down> (n div k) \<odot> k"
Pd_shuffle_to_top_range: fixes ds' :: "'b set"   and Y :: "'a set"   and d :: "'b" shows "Field (Pd_shuffle_to_top ds' Y d) \<subseteq> {x. Xd x = d}"
ccBind_eq: fixes v :: "Vars.var"   and e :: "exp"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and G :: "CoCalls" shows "ccBind v e\<cdot>(ae, G) = (if v--v\<notin>G \<or> \<not> isVal e then ccExp_bot_syn (ae v) e G|` fv e else fv e\<^sup>2)"
normalize_quot_swap: fixes a :: "'a"   and b :: "'a" assumes "a \<noteq> (0::'a)"   and "b \<noteq> (0::'a)" shows "normalize_quot (b, a) = (snd (normalize_quot (a, b)) div unit_factor (fst (normalize_quot (a, b))), fst (normalize_quot (a, b)) div unit_factor (fst (normalize_quot (a, b))))"
dropWhile_neq_rev: fixes xs :: "'c list"   and x :: "'c" assumes "distinct xs"   and "x \<in> set xs" shows "dropWhile (\<lambda>y. y \<noteq> x) (rev xs) = x # rev (takeWhile (\<lambda>y. y \<noteq> x) xs)"
simps_5: fixes r :: "'c rexp"   and s :: "'c rexp" shows "ACI_norm_alt (Plus r s) = ACI_nPlus (ACI_norm_alt r) (ACI_norm_alt s)"
dioph_memb_def: fixes m :: "int" shows "dioph_memb m = (\<lambda>(is, js). is ! int_encode m \<noteq> None)"
RedSFAss: fixes P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and F :: "char list"   and t :: "ty"   and D :: "char list"   and sh :: "sheap"   and sfs :: "sfields"   and i :: "init_state"   and sfs' :: "sfields"   and v :: "val"   and sh' :: "sheap"   and h :: "heap"   and l :: "sfields" assumes "P \<turnstile> C has F,Static:t in D"   and "sh D = \<lfloor>(sfs, i)\<rfloor>"   and "sfs' = sfs(F \<mapsto> v)"   and "sh' = sh(D \<mapsto> (sfs', i))" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sF{D} := Val v,(h, l, sh),True\<rangle> \<rightarrow> \<langle>unit,(h, l, sh'),False\<rangle>"
posk: fixes k :: "nat"   and x :: "'a" shows "0 \<le> f ((T ^^ k) x)"
is_expansion_X: shows "is_expansion (MS (MSLCons (1, 1) MSLNil) (\<lambda>x. x)) [\<lambda>x. x]"
index_mat_swaprows_1: fixes i :: "nat"   and A :: "'b mat"   and j :: "nat"   and k :: "nat"   and l :: "nat" assumes "i < dim_row A"   and "j < dim_col A" shows "swaprows k l A $$ (i, j) = (if k = i then A $$ (l, j) else if l = i then A $$ (k, j) else A $$ (i, j))"
FlexRayController_def: fixes return :: "nat \<Rightarrow> 'a Frame list"   and recv :: "nat \<Rightarrow> 'a Frame list"   and c :: "Config"   and store :: "nat \<Rightarrow> 'a Frame list"   and send :: "nat \<Rightarrow> 'a Frame list"   and get :: "nat \<Rightarrow> nat list" shows "FlexRayController return recv c store send get \<equiv> \<exists>activation. Scheduler c activation \<and> BusInterface activation return recv store send get"
induced_function_integral_aux_1: fixes A :: "'a set"   and f :: "'a \<Rightarrow> real" assumes "A \<in> sets M"   and "integrable M f"   and "\<And>x. 0 \<le> f x" shows "integrable M (induced_function A f)"
adj_sum: fixes x :: "'sz"   and y :: "'sz"   and \<nu>1 :: "(real, 'sz) vec"   and \<nu>2 :: "(real, 'sz) vec" shows "RSadj x y (\<nu>1 + \<nu>2) = RSadj x y \<nu>1 + RSadj x y \<nu>2"
cont_const: fixes c :: "'b" shows "cont (\<lambda>x. c)"
alw_iff_sdrop: fixes P :: "'d stream \<Rightarrow> bool"   and \<omega> :: "'d stream" shows "alw P \<omega> = (\<forall>m. P (sdrop m \<omega>))"
FGModuleEnd_avg_proj_right: fixes W :: "'v set"   and U :: "'v set" assumes "RModule UNIV (\<sharp>\<cdot>) W \<and> W \<subseteq> V"   and "RModule FG (\<cdot>) U \<and> U \<subseteq> V"   and "add_independentS [W, U]"   and "V = W \<oplus> U" shows "GRepEnd (((\<sharp>\<cdot>) ((1::'f) / ordG) \<circ> (\<Sum>g\<in>G. (*\<cdot>) (- g) \<circ> (\<lambda>a. if a \<in> (\<Oplus>G\<leftarrow>[W, U]. G) then W\<oplus>U\<leftarrow> a ! 1 else (0::'v)) \<circ> (*\<cdot>) g)) \<down> V)"
r: shows "\<forall>row_index\<in>I. row_index < n"
simps_6: fixes \<nu> :: "'a ltln"   and \<nu>' :: "'a ltln" shows "pure_universal (\<nu> R\<^sub>n \<nu>') = (\<nu> = false\<^sub>n \<or> pure_universal \<nu>')"
simps_3: fixes l :: "'a bdd"   and r :: "'a bdd"   and w :: "bool"   and ws :: "bool list"   and x :: "'a" shows "bddinsert (Branch l r) (w # ws) x = (if w then Branch l (bddinsert r ws x) else Branch (bddinsert l ws x) r)"
Plus_toplevel_summands: fixes r :: "'b rexp"   and s :: "'b rexp"   and t :: "'b rexp" assumes "Plus r s \<in> toplevel_summands t" shows "False"
T_insert3_def2: fixes x :: "'a"   and t :: "'a tree"   and n :: "nat" shows "T_insert3 x (t, n) = T_ins3 n 0 x t + 1"
drop_Cons': fixes n :: "nat"   and x :: "'b"   and xs :: "'b list" shows "drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)"
iteration_sep3: fixes y :: "'a"   and x :: "'a" assumes "y \<cdot> x \<le> x \<cdot> (x + y)" shows "(x + y)\<^sup>\<infinity> = x\<^sup>\<infinity> \<cdot> y\<^sup>\<infinity>"
step: fixes x :: "'a"   and A' :: "'a set"   and s :: "'b" assumes "x \<in> A'"   and "I A' s"   and "A' \<subseteq> A" shows "I (A' - {x}) (f x s)"
functor_axioms: shows "functor B.VVV.comp (\<cdot>\<^sub>B) B.HoHV"
isomorphic_to_zero_right: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and smult' :: "('b, 'a) aezfun \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'c \<Rightarrow> 'd"   and W :: "'d set" assumes "FGModuleIso G smult V smult' T W"   and "W = 0" shows "V = 0"
simps: fixes uu :: "'a extT" shows "uu\<^sup>\<star> = extT.Top"
comp_null_2: fixes f :: "'j" shows "f \<cdot>\<^sub>J J.null = J.null"
distinct_27: fixes x31 :: "poscheckvariant"   and x32 :: "('c, 'd) Term.term"   and x33 :: "('c, 'd) Term.term"   and x61 :: "poscheckvariant"   and x62 :: "('c, 'd) Term.term"   and x63 :: "('c, 'd) Term.term" shows "\<langle>x31: x32 \<doteq> x33\<rangle> \<noteq> \<langle>x61: x62 \<in> x63\<rangle>"
sum_incoming_alt_flow: fixes u :: "nat" assumes "u \<in> V" shows "sum f (incoming u) = (\<Sum>v\<in>V. f (v, u))"
cdot_square_norm: fixes x :: "'b" shows "x \<bullet>\<^sub>C x = complex_of_real ((norm x)\<^sup>2)"
comp_char: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" shows "B.VV.comp g f = (if B.VV.arr f \<and> B.VV.arr g \<and> B.VxV.seq g f then B.VxV.comp g f else B.VxV.null)"
simps: fixes a1 :: "pi"   and a2 :: "pi" shows "a1 \<sim> a2 = (\<exists>P Q. a1 = P \<and> a2 = Q \<and> P \<leadsto>[bisim] Q \<and> Q \<sim> P)"
add_cancel_right_right: fixes plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and a :: "'k"   and b :: "'k" assumes "class.cancel_comm_monoid_add plus minus zero" shows "(a = plus a b) = (b = zero)"
induced_def: fixes a :: "'a"   and b :: "'a" shows "(a \<lhd> b) = (b \<in> Orb a)"
cadd_le_self: fixes \<kappa> :: "V"   and a :: "V" assumes "Card \<kappa>" shows "\<kappa> \<le> \<kappa> \<oplus> a"
Ssc: fixes c :: "char list set"   and s1 :: "char list \<Rightarrow> int"   and s2 :: "char list \<Rightarrow> int" assumes "c \<subseteq> LVE"   and "s1 = s2 on \<Union> (SS ` c)" shows "S s1 = S s2 on c"
hyps_1: fixes t :: "('a, 'b) tm" assumes "n = num_args (fun t)"   and "is_App t" shows "chop t = emb_step_at (replicate (num_args (fun t)) dir.Left) dir.Right t"
f: shows "\<guillemotleft>MkArr (Dom a') (Cod a) (Map g) : a' \<Rightarrow> a\<guillemotright>"
simps_2: fixes xs :: "('a, 'b) Combinators list" shows "removeShadowRules1_alternative_rev (DenyAll # xs) = [DenyAll]"
map_prod_simp: fixes f :: "'m \<Rightarrow> 'k"   and g :: "'n \<Rightarrow> 'l"   and a :: "'m"   and b :: "'n" shows "map_prod f g (a, b) = (f a, g b)"
flip_def: fixes f :: "'h \<Rightarrow> 'g \<Rightarrow> 'i" shows "flip f = (\<lambda>b a. f a b)"
div_op_SOME: shows "is_div_op (\<lambda>a b. SOME k. k * b = a)"
outer_measure_empty: fixes M :: "'b set set"   and f :: "'b set \<Rightarrow> ennreal" assumes "positive M f"   and "{} \<in> M" shows "outer_measure M f {} = 0"
DJ_qf: fixes f :: "fm \<Rightarrow> fm" assumes "\<forall>p. qfree p \<longrightarrow> qfree (f p)" shows "\<forall>p. qfree p \<longrightarrow> qfree (DJ f p)"
that: fixes a :: "'a"   and \<pi> :: "'b \<Rightarrow> 'a" assumes "product_cone J (\<cdot>) D a \<pi>" shows "thesis"
listwordinterval_adjacent: fixes ss :: "('b word \<times> 'b word) list" shows "\<Union> (interval_of ` set (listwordinterval_adjacent ss)) = \<Union> (interval_of ` set ss)"
ticket_authentic: fixes B :: "agent"   and Tk :: "nat"   and A :: "agent"   and K :: "nat"   and evs :: "event list" assumes "Crypt (shrK B) \<lbrace>Number Tk, Agent A, Key K\<rbrace> \<in> parts (knows Spy evs)"   and "B \<notin> bad"   and "evs \<in> bankerb_gets" shows "Says Server A (Crypt (shrK A) \<lbrace>Number Tk, Agent B, Key K, Crypt (shrK B) \<lbrace>Number Tk, Agent A, Key K\<rbrace>\<rbrace>) \<in> set evs"
can_exec_cons: fixes root :: "(att \<times> char list, att, nat) env"   and x :: "operation"   and root' :: "(att \<times> char list, att, nat) env"   and xs :: "operation list" assumes "root \<midarrow>x\<rightarrow> root'"   and "can_exec root' xs" shows "can_exec root (x # xs)"
A_mb_k: shows "A $ mb $ mod_type_class.from_nat k \<noteq> (0::'a)"
Equiv_Exec_stable_def: fixes equiv_states :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and localState1 :: "'f \<Rightarrow> 'd"   and localState2 :: "'g \<Rightarrow> 'e"   and input_fun1 :: "'h \<Rightarrow> 'i message_af"   and input_fun2 :: "'h \<Rightarrow> 'j message_af"   and output_fun1 :: "'f \<Rightarrow> 'k message_af"   and output_fun2 :: "'g \<Rightarrow> 'k message_af"   and trans_fun1 :: "'i message_af \<Rightarrow> 'f \<Rightarrow> 'f"   and trans_fun2 :: "'j message_af \<Rightarrow> 'g \<Rightarrow> 'g"   and k1 :: "nat"   and k2 :: "nat"   and c1 :: "'f"   and c2 :: "'g" shows "Equiv_Exec_stable equiv_states localState1 localState2 input_fun1 input_fun2 output_fun1 output_fun2 trans_fun1 trans_fun2 k1 k2 c1 c2 \<equiv> \<forall>input m. Equiv_Exec m equiv_states localState1 localState2 input_fun1 input_fun2 output_fun1 output_fun2 trans_fun1 trans_fun2 k1 k2 (f_Exec_Comp trans_fun1 (map input_fun1 input \<odot> k1) c1) (f_Exec_Comp trans_fun2 (map input_fun2 input \<odot> k2) c2)"
fps_compose_fps_X: fixes a :: "'a fps" shows "a oo fps_X = a"
VPowE: fixes A :: "V"   and B :: "V" assumes "A \<in>\<^sub>\<circ> VPow B" obtains "A \<subseteq>\<^sub>\<circ> B"
frac_eq_0_iff: fixes x :: "'b" shows "(frac x = (0::'b)) = (x \<in> \<int>)"
simps_1: fixes ys :: "('f \<times> 'e) list" shows "compose [] ys = []"
mat_size: shows "M_mat signs subsets \<in> carrier_mat (length signs) (length signs)"
rMulLC: fixes R :: "('c, 'd) Ring_scheme"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "Ring R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "x \<cdot>\<^sub>r\<^bsub>R\<^esub> (y \<cdot>\<^sub>r\<^bsub>R\<^esub> z) = y \<cdot>\<^sub>r\<^bsub>R\<^esub> (x \<cdot>\<^sub>r\<^bsub>R\<^esub> z)"
map_map: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "list.map f (list.map g xs) = list.map (f \<circ> g) xs"
assms_4: shows "x\<^sub>1 \<le> x\<^sub>2"
VInter0I: fixes A :: "V" assumes "A = 0" shows "\<Inter>\<^sub>\<circ> A = 0"
lookup_trie_update: fixes ks :: "'e list"   and v :: "'d"   and t :: "('e, 'd) trie"   and ks' :: "'e list" shows "lookup_trie (trie_update ks v t) ks' = (if ks = ks' then Some v else lookup_trie t ks')"
HStartBallot_HInv5_q2: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HStartBallot s s' p"   and "p \<noteq> q"   and "\<exists>D\<in>MajoritySet. \<exists>qq. \<forall>d\<in>D. bal (dblock s q) < mbal (disk s d qq) \<and> \<not> hasRead s q d qq" shows "\<exists>D\<in>MajoritySet. \<exists>qq. \<forall>d\<in>D. bal (dblock s' q) < mbal (disk s' d qq) \<and> \<not> hasRead s' q d qq"
Ex1_Normal_form_part2: fixes k :: "int" shows "\<forall>f. (\<forall>n. f n = n + k) \<longrightarrow> f \<in> CofGroups.Ex1"
A: shows "[n] \<cdot> \<one> = (\<lambda>m. p_residue m (int n))"
path_defs_1: fixes g :: "real \<Rightarrow> 'a" shows "path g = continuous_on {0..1} g"
path_component_of_canon_iff: fixes T :: "'b set"   and a :: "'b"   and b :: "'b" shows "path_component_of (top_of_set T) a b = path_component T a b"
a_closure: fixes x :: "'a" shows "am_d (ad x) = ad x"
secgwext_host_attributes_def: shows "secgwext_host_attributes \<equiv> [''hypervisor'' \<mapsto> PolEnforcePoint, ''securevm1'' \<mapsto> DomainMember, ''securevm2'' \<mapsto> DomainMember, ''publicvm1'' \<mapsto> AccessibleMember, ''publicvm2'' \<mapsto> AccessibleMember]"
pred_mapping_le: fixes i :: "nat"   and P1 :: "'a \<Rightarrow> nat option"   and P2 :: "'a \<Rightarrow> nat option" assumes "pred_mapping ((\<le>) i) P1"   and "rel_mapping (\<le>) P1 P2" shows "pred_mapping ((\<le>) i) P2"
that: fixes Ks' :: "'a list" assumes "set (take (length Us) Ks) \<subseteq> set Ks'"   and "set Ks' \<subseteq> set (take (length Us) Ks) \<union> {\<zero>}"   and "length Ks' = length Us"   and "combine Ks' Us = a" shows "thesis"
distinct_list_of_dlist: fixes dxs :: "'a set_dlist" shows "distinct (list_of_dlist dxs)"