hcju/isabelleps huggingface dataset:
I: fixes n :: "nat" shows "\<integral>\<^sup>+ x. ennreal (norm (u (r n) x)) \<partial>M < ennreal ((1 / 2) ^ n)"
assms: shows "low_equal s1 s2 \<and> get_delayed_pool s1 = [] \<and> get_delayed_pool s2 = []"
that_3: fixes t\<^sub>1 :: "'a"   and t\<^sub>2 :: "'a" assumes "t = app t\<^sub>1 t\<^sub>2"   and "no_abs t\<^sub>1"   and "no_abs t\<^sub>2" shows "thesis"
u_1: shows "\<forall>n. u n \<in> f ` X"
ServiceCenter_a_l: fixes i :: "nat \<Rightarrow> ECall_Info list"   and a :: "nat \<Rightarrow> aType list"   and t :: "nat" assumes "ServiceCenter i a" shows "length (a t) \<le> Suc 0"
rep_eq: fixes x :: "Arity"   and xa :: "Arity" shows "(x \<sqsubseteq> xa) = (Rep_Arity xa \<le> Rep_Arity x)"
steps_add: fixes c :: "nat \<times> cell list \<times> cell list"   and p :: "(action \<times> nat) list \<times> nat"   and m :: "nat"   and n :: "nat" shows "steps c p (m + n) = steps (steps c p m) p n"
valid_distinguishing_formula: fixes P' :: "'state" shows "P' \<Turnstile> distinguishing_weak_formula P Q = (\<forall>x\<in>{- p \<bullet> (SOME x. weak_formula x \<and> supp x \<subseteq> supp (p \<bullet> P) \<and> x distinguishes (p \<bullet> P) from (p \<bullet> Q)) |p. True}. P' \<Turnstile> x)"
extr_insert_Chan: fixes bad :: "agent set"   and IK :: "msg set"   and c :: "tag"   and A :: "agent"   and B :: "agent"   and M :: "msg"   and H :: "chan set" shows "extr bad IK (insert (Chan c A B M) H) = (if c = insec \<or> c = auth \<or> A \<in> bad \<or> B \<in> bad then insert M (extr bad IK H) else extr bad IK H)"
stream_enc_cut_same: fixes I :: "(nat + nat set) list"   and w :: "'a list" assumes "\<forall>x\<in>set I. case x of Inl a \<Rightarrow> True | Inr P \<Rightarrow> finite P" shows "stream_enc (w, I) = cut_same (any, replicate (length I) False) (stream_enc (w, I)) @- sconst (any, replicate (length I) False)"
mycases_def: fixes x :: "'a"   and y :: "'a" shows "mycases x y = Plus (mycasexxy x y) (Plus (mycasexyx x y) (Plus (mycasexx x y) (Plus (mycasexy x y) (question (mycasex x y)))))"
reduce_wf_body_to_stmts: fixes p :: "prog"   and s\<^sub>L :: "stmt"   and s\<^sub>H :: "stmt" assumes "body_split p s\<^sub>L s\<^sub>H" shows "wfs \<Gamma>\<^sub>0 (body p) = (wfs \<Gamma>\<^sub>0 s\<^sub>L \<and> wfs \<Gamma>\<^sub>0 s\<^sub>H)"
slice_kind_only_one_True_edge: fixes a :: "'edge"   and a' :: "'edge"   and S :: "'node SDG_node set" assumes "sourcenode a = sourcenode a'"   and "targetnode a \<noteq> targetnode a'"   and "valid_edge a"   and "valid_edge a'"   and "intra_kind (kind a)"   and "intra_kind (kind a')"   and "slice_kind S a = (\<lambda>s. True)\<^sub>\<surd>" shows "slice_kind S a' = (\<lambda>s. False)\<^sub>\<surd>"
rec_le_def: shows "rec_le = CN rec_disj [rec_less, rec_eq]"
list_all2_all_nthI: fixes a :: "'b list"   and b :: "'c list"   and P :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "length a = length b"   and "\<And>n. n < length a \<Longrightarrow> P (a ! n) (b ! n)" shows "list_all2 P a b"
weakPsiCongScopeExt: fixes x :: "name"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "x \<sharp> \<Psi>"   and "x \<sharp> P" shows "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>(P \<parallel> Q) \<doteq> P \<parallel> \<lparr>\<nu>x\<rparr>Q"
callee_of_core_def: fixes core :: "('a, 'b, 'c, 'd, 'e, 'f) core" shows "callee_of_core core = map_fun id (map_fun id (map_spmf (Pair ()))) (cpoke core) \<oplus>\<^sub>O cfunc_adv core \<oplus>\<^sub>O cfunc_usr core"
snd_scaleR: fixes r :: "real"   and A :: "'a \<times> 'b" shows "snd (r *\<^sub>R A) = r *\<^sub>R snd A"
simps_3: fixes n :: "nat"   and s :: "'state_t"   and execs :: "'dom_t \<Rightarrow> 'action_t list list" assumes "interrupt (Suc n)" shows "run (Suc n) (Some s) execs = run n (Some (cswitch (Suc n) s)) execs"
divide_pos_neg: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero x"   and "less y zero" shows "less (divide x y) zero"
simps_3: fixes h :: "'b tree" shows "U Del_min [h] = 2 * log 2 (real (size h + 1)) + 5"
l9_4_2_aux: fixes S :: "'p"   and C :: "'p"   and R :: "'p"   and A :: "'p"   and P :: "'p"   and Q :: "'p"   and U :: "'p"   and V :: "'p" assumes "S C Le R A"   and "P Q TS A C"   and "Col R P Q"   and "P Q Perp A R"   and "Col S P Q"   and "P Q Perp C S"   and "R Out U A"   and "S Out V C" shows "P Q TS U V"
nat_sym_cases: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "\<And>a b. a < b \<Longrightarrow> P a b"   and "\<And>a. P a a"   and "\<And>a b. P b a \<Longrightarrow> P a b" shows "P a b"
isRes_def: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and prot :: "'c \<Rightarrow> bool"   and protFw :: "'c \<Rightarrow> bool"   and \<sigma> :: "'c" assumes "ProblemIkTpart wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw" shows "ProblemIkTpart.isRes wtFsym resOf \<sigma> \<equiv> \<exists>f. wtFsym f \<and> resOf f = \<sigma>"
d_i_2: shows "rat_of_int (d fs' i) = \<parallel>gs2.gso (i - 1)\<parallel>\<^sup>2 / \<parallel>fs.gs.gso (i - 1)\<parallel>\<^sup>2 * rat_of_int (d fs i)"
boundedE: fixes A :: "'a word set"   and x :: "'a word" assumes "finite A"   and "A \<noteq> {}"   and "signed.Max A \<le>s x" obtains "\<And>a. a \<in> A \<Longrightarrow> a \<le>s x"
Cons_1: fixes i :: "nat"   and c :: "bool"   and y :: "nat" shows "fst (fold_map_idx f i c xs) = (c \<or> foldl g y (snd (fold_map_idx f i c xs)) < foldl g y xs)"
nonempty_FIN_wrt_impl_CP: fixes U :: "(nat \<Rightarrow> nat option) set"   and \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option" assumes "U \<noteq> {}"   and "U \<in> FIN_wrt \<psi>" shows "U \<in> CP_wrt \<psi>"
cont_case_prodD1: fixes luba :: "'b set \<Rightarrow> 'b"   and lubb :: "'c set \<Rightarrow> 'c"   and orda :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ordb :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and lubc :: "'d set \<Rightarrow> 'd"   and ordc :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and x :: "'b" assumes "cont (prod_lub luba lubb) (rel_prod orda ordb) lubc ordc (\<lambda>(x, y). f x y)"   and "class.preorder orda (mk_less orda)"   and "lub_singleton luba" shows "cont lubb ordb lubc ordc (f x)"
conv_complement_sub_inf: fixes x :: "'a"   and y :: "'a" shows "id x \<sqinter> - (x \<sqinter> y) \<sqinter> y = bot"
inf_greatest: fixes A :: "'a set"   and z :: "'a" assumes "A \<subseteq> carrier (fpl L f)"   and "z \<in> carrier (fpl L f)"   and "\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq>\<^bsub>fpl L f\<^esub> x" shows "z \<sqsubseteq>\<^bsub>fpl L f\<^esub> \<Sqinter>\<^bsub>fpl L f\<^esub>A"
b3: shows "(a \<rightleftharpoons> b) \<bullet> nat_of = nat_of"
IdNatTrans_map: fixes X :: "'g"   and F :: "('g, 'j, 'h, 'k, 'i, 'l) Functor" assumes "X \<in> obj\<^bsub>CatDom F\<^esub>" shows "IdNatTrans F $$ X = id\<^bsub>CatCod F\<^esub> (F @@ X)"
subset_int_mono: fixes A :: "'d set"   and B :: "'d set"   and C :: "'d set" assumes "A \<subseteq> B" shows "A \<inter> C \<subseteq> B \<inter> C"
WhileBodyThrow: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and c :: "expr"   and e' :: "expr"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" assumes "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>true,s\<^sub>1\<rangle>"   and "P,E \<turnstile> \<langle>c,s\<^sub>1\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>2\<rangle>" shows "P,E \<turnstile> \<langle>while (e) c,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>2\<rangle>"
diff_def: fixes u :: "nat"   and v :: "nat" shows "diff u v = f v (c_pair u v)"
simps_1: fixes M :: "nat" shows "remove_weights (Input M) = Input M"
lfp_unfold: fixes f :: "'a \<Rightarrow> 'a" assumes "mono f" shows "complete_lattice_class.lfp f = f (complete_lattice_class.lfp f)"
K2_centre_on_equator_meridian_2: shows "proj2_incident K2_centre meridian"
no_head_gen_code_ind_step: fixes vs :: "'a list list"   and b :: "'a list"   and us :: "'a list list"   and ku :: "nat"   and kv :: "nat" assumes "vs \<in> lists (no_head_gen \<C> b)"   and "us \<in> lists (no_head_gen \<C> b)"   and "b \<in> \<C>"   and "[b] \<^sup>@ ku \<cdot> Ref \<C> us = [b] \<^sup>@ kv \<cdot> Ref \<C> vs" shows "ku = kv"
hyps: shows "P \<turnstile> typeof\<^bsub>h\<^esub> v = \<lfloor>T\<rfloor>"
filtered_def: fixes X :: "'a set" shows "filtered X = (\<forall>Y. finite Y \<and> Y \<subseteq> X \<longrightarrow> (\<exists>x\<in>X. \<forall>y\<in>Y. x \<le> y))"
intro: fixes A :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sim :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>a b a'. \<lbrakk>A a b; sim a a'\<rbrakk> \<Longrightarrow> \<exists>b'. B a' b' \<and> sim b b'" shows "Simulation A B sim"
hom_from_subgroup_generated: fixes h :: "'a \<Rightarrow> 'b"   and H :: "('b, 'c) monoid_scheme"   and A :: "'a set" assumes "h \<in> hom B H" shows "h \<in> hom (subgroup_generated B A) H"
prems_1: shows "P \<turnstile> \<langle>e # es,s\<rangle> [\<Rightarrow>] \<langle>es',s'\<rangle>"
succs_is_node: fixes x :: "'a" assumes "is_node x" shows "list_all is_node (succs x)"
cases: fixes x :: "('a, 'b) L_action" obtains \<alpha> :: "'a" where "x = L_action.Act \<alpha>"   | uu_ :: "'b" where "x = Eff uu_"
recognises_cons_step: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and h :: "String.literal \<times> value list"   and t :: "(String.literal \<times> value list) list" assumes "recognises_execution e s r (h # t)" shows "step e s r (fst h) (snd h) \<noteq> None"
surj_pair: fixes p :: "'h \<times> 'i" shows "\<exists>x y. p = (x, y)"
Deriv_pderiv: fixes c :: "'a"   and r :: "'a rexp" shows "Deriv c (lang r) = \<Union> (lang ` pderiv c r)"
hom_add: fixes x :: "'c poly"   and y :: "'c poly" shows "poly_lift (x + y) = poly_lift x + poly_lift y"
cring_simprules_28: fixes R :: "('b, 'c) ring_scheme"   and x :: "'b"   and y :: "'b" assumes "cring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "\<ominus>\<^bsub>R\<^esub> x \<otimes>\<^bsub>R\<^esub> y = \<ominus>\<^bsub>R\<^esub> (x \<otimes>\<^bsub>R\<^esub> y)"
pointermap_sane_appendD: fixes s :: "('a, 'b) pointermap_scheme"   and m :: "'a" assumes "pointermap_sane s"   and "m \<notin> set (entries s)" shows "pointermap_sane (pointermap_insert m s)"
symKey_compromise: fixes evs :: "event list" assumes "evs \<in> set_mr" shows "\<forall>SK KK. SK \<in> symKeys \<longrightarrow> (\<forall>K\<in>KK. K \<notin> range priEK) \<longrightarrow> (Key SK \<in> analz (Key ` KK \<union> knows Spy evs)) = (SK \<in> KK \<or> Key SK \<in> analz (knows Spy evs))"
pdevs_apply_pdevs_of_list: fixes xs :: "'a list"   and i :: "nat" shows "pdevs_apply (pdevs_of_list xs) i = (if i < length xs then xs ! i else (0::'a))"
access_matrix_pretty_ipv6_code_def: fixes c :: "parts_connection"   and rs :: "128 simple_rule list" shows "access_matrix_pretty_ipv6_code c rs \<equiv> if \<not> simple_firewall_without_interfaces rs then undefined else let W = build_ip_partition c rs; R = map getOneIp W; U = all_pairs R in (zip (map ipv6addr_toString R) (map ipv6addr_wordinterval_toString W), map (\<lambda>(x, y). (ipv6addr_toString x, ipv6addr_toString y)) (filter (\<lambda>(s, d). runFw s d c rs = Decision FinalAllow) (all_pairs R)))"
nxt_holds_iff_snth: fixes y :: "nat"   and P :: "'b \<Rightarrow> bool"   and xs :: "'b stream" shows "(nxt ^^ y) (holds P) xs = P (xs !! y)"
valid_bool_split: fixes \<tau> :: "'a state \<times> 'a state"   and A :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "(\<tau> \<Turnstile> \<upsilon> A) = (\<tau> \<Turnstile> A \<triangleq> null \<or> \<tau> \<Turnstile> A \<or> \<tau> \<Turnstile> not A)"
seq_doesnt_read: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and x :: "'Var"   and c' :: "('Var, 'AExp, 'BExp) Stmt" assumes "doesnt_read c x" shows "doesnt_read (c ;; c') x"
assms_1: shows "(\<^bold>@ a p) at b in (ps, a) # branch"
split: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "'c"   and f2 :: "'d tree \<Rightarrow> 'd \<Rightarrow> 'd tree \<Rightarrow> 'c"   and tree :: "'d tree" shows "P (case tree of \<langle>\<rangle> \<Rightarrow> f1 | \<langle>x, xa, xb\<rangle> \<Rightarrow> f2 x xa xb) = ((tree = \<langle>\<rangle> \<longrightarrow> P f1) \<and> (\<forall>x21 x22 x23. tree = \<langle>x21, x22, x23\<rangle> \<longrightarrow> P (f2 x21 x22 x23)))"
subst_sound_scl: fixes P :: "'s list"   and CAs :: "'a literal multiset list"   and I :: "'a set"   and \<eta> :: "'s" assumes "length P = length CAs"   and "\<And>\<sigma>. is_ground_subst \<sigma> \<Longrightarrow> I \<Turnstile>m mset CAs \<cdot>cm \<sigma>"   and "is_ground_subst \<eta>" shows "I \<Turnstile>m mset (CAs \<cdot>\<cdot>cl P) \<cdot>cm \<eta>"
sg1: shows "lose t = [True] \<or> lose t = [False]"
index_map_vec_1: fixes i :: "nat"   and v :: "'a Matrix.vec"   and f :: "'a \<Rightarrow> 'b" assumes "i < dim_vec v" shows "map_vec f v $v i = f (v $v i)"
P4: shows "\<lfloor>\<lambda>w. \<forall>x. Dependent x w \<longrightarrow> (\<exists>xa. (Independent xa \<^bold>\<and> x dependsOn xa) w)\<rfloor>"
Bex_set_list_ex: fixes xs :: "'g list"   and P :: "'g \<Rightarrow> bool" shows "Bex (set xs) P = list_ex P xs"
chainTransitionAppend_2: fixes P :: "pi"   and \<alpha> :: "freeRes"   and P'' :: "pi"   and P' :: "pi" assumes "P \<Longrightarrow>\<alpha> \<prec> P''"   and "P'' \<Longrightarrow>\<^sub>\<tau> P'" shows "P \<Longrightarrow>\<alpha> \<prec> P'"
smult_r_distr: fixes a :: "'a"   and x :: "'c"   and y :: "'c" assumes "a \<in> carrier K"   and "x \<in> carrier V"   and "y \<in> carrier V" shows "a \<odot>\<^bsub>V\<^esub> (x \<oplus>\<^bsub>V\<^esub> y) = a \<odot>\<^bsub>V\<^esub> x \<oplus>\<^bsub>V\<^esub> a \<odot>\<^bsub>V\<^esub> y"
restrict_relation_of_weak_ranking_Cons: fixes A :: "'b set"   and As :: "'b set list" assumes "is_weak_ranking (A # As)" shows "restrict_relation (\<Union> (set As)) (of_weak_ranking (A # As)) = of_weak_ranking As"
arr_eval: fixes f :: "'c local.term" assumes "Arr f" shows "D.arr \<lbrace>f\<rbrace>"
Lunit_in_lunit: fixes f :: "'c local.term set"   and t :: "'c local.term" assumes "\<F>C.arr f"   and "t \<in> f" shows "\<^bold>\<l>\<^bold>[t\<^bold>] \<in> \<l>[f]"
finterI: fixes c :: "'a"   and A :: "'a fset"   and B :: "'a fset" assumes "c |\<in>| A"   and "c |\<in>| B" shows "c |\<in>| A |\<inter>| B"
sum_list_zeroI': fixes xs :: "'c list" assumes "\<And>i. i < length xs \<Longrightarrow> xs ! i = (0::'c)" shows "sum_list xs = (0::'c)"
strict_implies_order: fixes b :: "'e"   and a :: "'e" assumes "b < a" shows "b \<le> a"
eq1: shows "(lp q - lp p) \<oplus> lt p = lt q"
vdiff_vsingleton: fixes a :: "V"   and A :: "V" assumes "a \<notin>\<^sub>\<circ> A" shows "A -\<^sub>\<circ> set {a} = A"
preserve_order_def: fixes x :: "'a"   and y :: "'a"   and u :: "'b"   and v :: "'b" shows "preserve_order x y u v \<equiv> (x < y \<longrightarrow> u < v) \<and> (y < x \<longrightarrow> v < u)"
em_assoc: shows "Sup \<circ> \<P> Sup = Sup \<circ> \<mu>"
triangle_contour_integrals_starlike_primitive: fixes S :: "complex set"   and f :: "complex \<Rightarrow> complex"   and a :: "complex"   and x :: "complex" assumes "continuous_on S f"   and "a \<in> S"   and "open S"   and "x \<in> S"   and "\<And>y. y \<in> S \<Longrightarrow> closed_segment a y \<subseteq> S"   and "\<And>b c. closed_segment b c \<subseteq> S \<Longrightarrow> contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c a) f = 0" shows "((\<lambda>x. contour_integral (linepath a x) f) has_field_derivative f x) (at x)"
set_minus_filter_out: fixes xs :: "'k list"   and y :: "'k" shows "set xs - {y} = set (filter (\<lambda>x. x \<noteq> y) xs)"
map_add_left_commute: fixes h\<^sub>0 :: "'c \<Rightarrow> 'd option"   and h\<^sub>1 :: "'c \<Rightarrow> 'd option"   and h\<^sub>2 :: "'c \<Rightarrow> 'd option" assumes "h\<^sub>0 \<bottom> h\<^sub>1" shows "h\<^sub>0 ++ (h\<^sub>1 ++ h\<^sub>2) = h\<^sub>1 ++ (h\<^sub>0 ++ h\<^sub>2)"
ce_rel_lm_34: fixes r :: "(nat \<times> nat) set"   and s :: "(nat \<times> nat) set" shows "ce_rel_to_set (r \<inter> s) = ce_rel_to_set r \<inter> ce_rel_to_set s"
select_convs_1: fixes view :: "'d \<Longrightarrow> 'e"   and coview :: "'f \<Longrightarrow> 'e"   and more :: "'g" shows "\<V>\<^bsub>\<lparr>view = view, coview = coview, \<dots> = more\<rparr>\<^esub> = view"
rtrancl_while_def: shows "local.rtrancl_while = while_option local.rtrancl_while_test local.rtrancl_while_step ([x], {x})"
fsubset_trans: fixes A :: "'a fset"   and B :: "'a fset"   and C :: "'a fset" assumes "A |\<subseteq>| B"   and "B |\<subseteq>| C" shows "A |\<subseteq>| C"
nat_pow_zero: fixes n :: "nat" assumes "n \<noteq> 0" shows "(0::'b) ^ n = (0::'b)"
augment_edge_rev_other: fixes u :: "nat"   and v :: "nat"   and e' :: "nat \<times> nat"   and f :: "nat \<times> nat \<Rightarrow> 'capacity"   and \<Delta> :: "'capacity" assumes "(u, v) \<notin> E"   and "e' \<noteq> (v, u)" shows "augment_edge f (u, v) \<Delta> e' = f e'"
set_eqE2: fixes R :: "('d, 'e) eq_object_scheme"   and A :: "'d set"   and B :: "'d set" assumes "A {.=}\<^bsub>R\<^esub> B" obtains "\<forall>a\<in>A. \<exists>b\<in>B. a .=\<^bsub>R\<^esub> b" and "\<forall>b\<in>B. \<exists>a\<in>A. b .=\<^bsub>R\<^esub> a"
f'_def: shows "f' = (\<lambda>x. f x - eval C x * b x powr e)"
exhaust_sel: fixes SkewBinomialTree :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" obtains "SkewBinomialTree = SkewBinomialHeapStruc.Node (SkewBinomialHeapStruc.val SkewBinomialTree) (SkewBinomialHeapStruc.prio SkewBinomialTree) (SkewBinomialHeapStruc.rank SkewBinomialTree) (SkewBinomialHeapStruc.children SkewBinomialTree)"
ENR_closedin_Un_local: fixes S :: "'a set"   and T :: "'a set" assumes "ENR S"   and "ENR T"   and "ENR (S \<inter> T)"   and "closedin (top_of_set (S \<union> T)) S"   and "closedin (top_of_set (S \<union> T)) T" shows "ENR (S \<union> T)"
AE_cong: fixes M :: "'b measure"   and P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "\<And>x. x \<in> space M \<Longrightarrow> P x = Q x" shows "almost_everywhere M P = almost_everywhere M Q"
is_empty_def: fixes A :: "'a pred" shows "is_empty A = (A = \<bottom>)"
simps_1: fixes a :: "floatarith"   and b :: "floatarith"   and vs :: "real list" shows "interpret_floatarith (Add a b) vs = interpret_floatarith a vs + interpret_floatarith b vs"
that: fixes u :: "'a \<Rightarrow> 'b" assumes "u \<in> [M]\<^bsub>v,n\<^esub>"   and "u c2 = - r" shows "thesis"
lex: shows "llexord r (lappend xs ys) (lappend xs zs)"
fps_inverse_minus: fixes f :: "'a fps" shows "inverse (- f) = - inverse f"
mult_monos_2: fixes a :: "'a"   and c :: "'a"   and d :: "'a"   and b :: "'a" assumes "a \<le> c"   and "d \<le> b"   and "a \<le> (0::'a)"   and "(0::'a) \<le> d" shows "a * b \<le> c * d"
map_add_find_right: fixes n :: "'f \<Rightarrow> 'e option"   and k :: "'f"   and xx :: "'e"   and m :: "'f \<Rightarrow> 'e option" assumes "n k = \<lfloor>xx\<rfloor>" shows "(m ++ n) k = \<lfloor>xx\<rfloor>"
isval_var_inv: fixes x :: "nat"   and P :: "bool" assumes "isval (EVar x)" shows "P"
inner_not_same_Basis: fixes u :: "'a"   and v :: "'a" assumes "u \<in> Basis"   and "v \<in> Basis"   and "u \<noteq> v" shows "u \<bullet> v = 0"
translation_def: shows "translation = (\<lambda>a\<in>M. restrict ((\<cdot>) a) M)"
append_Nil2: fixes xs :: "'b list" shows "xs \<cdot> \<epsilon> = xs"
RB: fixes x :: "'M"   and y :: "'M" assumes "x \<in> S"   and "y \<in> S"   and "norm (y - x) < R x" shows "norm (f y - f x) \<le> B * norm (y - x)"
ns_mul_ext_trans: fixes s :: "('a \<times> 'a) set"   and ns :: "('a \<times> 'a) set"   and A :: "'a multiset"   and B :: "'a multiset"   and C :: "'a multiset" assumes "trans s"   and "trans ns"   and "compatible_l ns s"   and "compatible_r ns s"   and "refl ns"   and "(A, B) \<in> ns_mul_ext ns s"   and "(B, C) \<in> ns_mul_ext ns s" shows "(A, C) \<in> ns_mul_ext ns s"
PO_s0g_secrecy_init: shows "init s0g \<subseteq> s0g_secrecy"
rep_eq: fixes x :: "float" shows "int_floor_fl x = \<lfloor>real_of_float x\<rfloor>"
denote_dim_pdo: fixes S :: "com"   and \<rho> :: "complex mat" assumes "well_com S"   and "\<rho> \<in> carrier_mat d d"   and "partial_density_operator \<rho>" shows "denote S \<rho> \<in> carrier_mat d d \<and> partial_density_operator (denote S \<rho>)"
circuit_in_def: fixes \<E> :: "'a set"   and X :: "'a set" shows "\<E>.circuit_in \<E> X = indep_system.circuit \<E> (\<E>.indep_in \<E>) X"
simps_1: shows "foldli [] c f \<sigma> = \<sigma>"
ccSp_Nil: fixes n :: "'node SDG_node" assumes "valid_SDG_node n" shows "n cc-[]\<rightarrow>\<^sub>d* n"
goodPar_def: fixes P :: "('a, 'b, 'varSort, 'var, 'c) param" shows "goodPar P \<equiv> Ball (termsOf P) good \<and> Ball (absOf P) goodAbs \<and> Ball (envsOf P) goodEnv"
fbrelation_vinsertI: fixes r :: "V"   and a :: "V"   and b :: "V" assumes "fbrelation r" shows "fbrelation (vinsert [a, b]\<^sub>\<circ> r)"
inj_mult: fixes x :: "nat"   and q :: "nat"   and y :: "nat"   and y' :: "nat" assumes "coprime x q"   and "y < q"   and "y' < q"   and "x * y mod q = x * y' mod q" shows "y = y'"
euler_mascheroni_sum_real: shows "(\<lambda>n. inverse (real (n + 1)) + ln (real (n + 1)) - ln (real (n + 2))) sums euler_mascheroni"
insertion_power: fixes f :: "nat \<Rightarrow> 'a"   and p :: "'a mpoly"   and n :: "nat" shows "insertion f (p ^ n) = insertion f p ^ n"
simps_1: fixes M :: "('a, 'b) message set"   and v :: "('a, 'b option) insert_message list" shows "is_certified_associated_string M (Inr v) = is_associated_string M v"
incl: fixes b :: "'a set" assumes "b \<in> B" shows "b \<subseteq> A"
is_lub_below_iff: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b set"   and x :: "'b"   and u :: "'b" assumes "class.po below"   and "po.is_lub below S x" shows "below x u = po.is_ub below S u"
pow_eq: fixes l :: "nat"   and m :: "nat" assumes "l = m" shows "x ^ l = x ^ m"
simps: fixes atom_ordering :: "('a \<times> 'a) set"   and R :: "'a Literal set \<Rightarrow> 'a Literal set \<Rightarrow> 'a Literal set \<Rightarrow> bool"   and S :: "'a Literal set set" assumes "propositional_atoms atom_ordering" shows "propositional_atoms.add_all_deducible_clauses R S = S \<union> propositional_atoms.all_deducible_clauses R S"
Rxa_nonzero: fixes a :: "'a" assumes "a \<in> carrier R"   and "a \<noteq> \<zero>" shows "R \<diamondsuit>\<^sub>p a \<noteq> {\<zero>}"
vcomp_vid_on_rel: fixes A :: "V"   and r :: "V" shows "vid_on A \<circ>\<^sub>\<circ> r = r \<restriction>\<^sup>r\<^sub>\<circ> A"
rep_eq: fixes x :: "real"   and xa :: "'c \<Rightarrow>\<^sub>C 'd" shows "apply_bcontfun (x *\<^sub>R xa) = (\<lambda>xb. x *\<^sub>R apply_bcontfun xa xb)"