hcju/isabelleps huggingface dataset:
leofI: fixes m :: "'b nres"   and m' :: "'b nres" assumes "nofail m \<Longrightarrow> m \<le> m'" shows "m \<le>\<^sub>n m'"
zero_component: fixes rep_nat_term :: "'a \<Rightarrow> (nat, nat) pp \<times> nat"   and splus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and is_scalar :: "'a itself \<Rightarrow> bool"   and lex_comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and deg_comp :: "('a \<Rightarrow> 'a \<Rightarrow> order) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> order"   and pot_comp :: "('a \<Rightarrow> 'a \<Rightarrow> order) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> order" assumes "class.nat_term_compare rep_nat_term splus less_eq less is_scalar lex_comp deg_comp pot_comp" shows "\<exists>x. snd (rep_nat_term x) = 0"
continuous_map_real_mult_left_eq: fixes X :: "'b topology"   and c :: "real"   and f :: "'b \<Rightarrow> real" shows "continuous_map X euclideanreal (\<lambda>x. c * f x) = (c = 0 \<or> continuous_map X euclideanreal f)"
assms_4: shows "\<^bold>g [^] z1 \<otimes> g' [^] z1' = a \<otimes> h [^] e"
soundnessForUNSAT: fixes state0 :: "State"   and F0 :: "Literal list list"   and state :: "State"   and decisionVars :: "nat set" assumes "isInitialState state0 F0"   and "(state0, state) \<in> transitionRelation F0 decisionVars"   and "getConflictFlag state = True"   and "getC state = []" shows "\<not> satisfiable F0"
is_unit_div_mult_cancel_right: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "a \<noteq> zero"   and "dvd.dvd times b one" shows "divide a (times b a) = divide one b"
postrec_ip1: shows "postrecording_event (swap_events i j t) (i + 1)"
from_nat_mono: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and one :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Rep :: "'d \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'd"   and i :: "nat"   and j :: "nat" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "i < j"   and "j < CARD('d)" shows "less (mod_type.from_nat Abs i) (mod_type.from_nat Abs j)"
mem_add_set: fixes A :: "rat set"   and B :: "rat set" assumes "Dedekind_Real.cut A"   and "Dedekind_Real.cut B" shows "Dedekind_Real.cut (add_set A B)"
test_shunting_bot: fixes x :: "'a"   and y :: "'a" assumes "test x"   and "test y" shows "(x \<le> y) = (x \<sqinter> ! y \<le> bot)"
Image_empty2: fixes R :: "('b \<times> 'a) set" shows "R `` {} = {}"
split_req_2: fixes xs :: "('a btree \<times> 'a) list"   and p :: "'a"   and ls :: "('a btree \<times> 'a) list"   and sub :: "'a btree"   and sep :: "'a"   and rs :: "('a btree \<times> 'a) list" assumes "split xs p = (ls @ [(sub, sep)], rs)"   and "sorted_less (separators xs)" shows "sep < p"
neg_eq_simps_2: fixes x :: "tr" assumes "neg\<cdot>x = FF" shows "x = TT"
split_if: fixes ks :: "'a list"   and ps :: "'a list"   and qs :: "'a list"   and ks' :: "'a list"   and ps' :: "'a list" assumes "split ks ps = (qs, ks', ps')" shows "ks = qs @ ks' \<and> ps = qs @ ps' \<and> (ks' \<noteq> [] \<and> ps' \<noteq> [] \<longrightarrow> hd ks' \<noteq> hd ps')"
psubstEnv_updEnv: fixes rho' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and x :: "'var"   and X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and xs :: "'varSort" shows "rho [x \<leftarrow> X]_xs &[rho'] = rho &[rho'] [x \<leftarrow> X #[rho']]_xs"
bool_to_real_def: fixes b :: "bool" shows "bool_to_real b = (if b then 1 else 0)"
rep_eq: fixes x :: "nat" shows "mapping_of (Var x) = Var\<^sub>0 x"
Spy_not_see_MS: fixes A :: "agent"   and B :: "agent"   and PMS :: "nat"   and evs :: "event list"   and NA :: "nat"   and NB :: "nat" assumes "Notes A \<lbrace>Agent B, Nonce PMS\<rbrace> \<in> set evs"   and "evs \<in> tls"   and "A \<notin> bad"   and "B \<notin> bad" shows "Nonce (PRF (PMS, NA, NB)) \<notin> analz (knows Spy evs)"
as1: shows "x mod (m * k) = y * k mod (m * k)"
rv_Low_Use_Low: fixes m :: "'node"   and as :: "'edge list"   and as' :: "'edge list"   and S :: "'node SDG_node set"   and cf :: "'var \<Rightarrow> 'val option"   and cf' :: "'var \<Rightarrow> 'val option" assumes "m -as\<rightarrow>\<^sub>\<surd>* (_Low_)"   and "m -as'\<rightarrow>\<^sub>\<surd>* (_Low_)"   and "get_proc m = Main"   and "\<forall>V\<in>rv S (CFG_node m). cf V = cf' V"   and "preds (slice_kinds S as) [(cf, undefined)]"   and "preds (slice_kinds S as') [(cf', undefined)]"   and "CFG_node (_Low_) \<in> S" shows "\<forall>V\<in>Use (_Low_). state_val (transfers (slice_kinds S as) [(cf, undefined)]) V = state_val (transfers (slice_kinds S as') [(cf', undefined)]) V"
restrict_restrict_spmf: fixes p :: "'c spmf"   and A :: "'c set"   and B :: "'c set" shows "p \<upharpoonleft> A \<upharpoonleft> B = p \<upharpoonleft> (A \<inter> B)"
eSuc_ile_mono: fixes n :: "enat"   and m :: "enat" shows "(eSuc n \<le> eSuc m) = (n \<le> m)"
wp_upred_def: fixes Q :: "('b, 'c) urel"   and r :: "'c upred" shows "Q wp r = \<lfloor>\<not> Q ;; (\<not> \<lceil>r\<rceil>\<^sub><)\<rfloor>\<^sub><"
path_via_C: 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 C :: "char list"   and Cs :: "char list list"   and wf_md :: "(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 \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool" assumes "P \<turnstile> Path C to C via Cs "   and "wf_prog wf_md P" shows "Cs = [C]"
Inf_greatest: fixes Inf :: "'a::type set \<Rightarrow> 'a::type"   and Sup :: "'a::type set \<Rightarrow> 'a::type"   and inf :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type"   and less_eq :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> bool"   and less :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> bool"   and sup :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type"   and bot :: "'a::type"   and top :: "'a::type"   and A :: "'a::type set"   and z :: "'a::type" assumes "class.complete_lattice (Inf::'a::type set \<Rightarrow> 'a::type) (Sup::'a::type set \<Rightarrow> 'a::type) (inf::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type) (less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (less::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (sup::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type) (bot::'a::type) (top::'a::type)"   and "\<And>x::'a::type. x \<in> (A::'a::type set) \<Longrightarrow> (less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (z::'a::type) x" shows "(less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (z::'a::type) ((Inf::'a::type set \<Rightarrow> 'a::type) (A::'a::type set))"
add_rel_0': fixes a :: "('a set \<times> 'b) set" assumes "a \<in> carrier_direct_lim" shows "add_rel a \<lfloor>U, \<zero>\<^bsub>U\<^esub>\<rfloor> = a"
mult_mat_vec_def: fixes A :: "'a mat"   and v :: "'a vec" shows "A *\<^sub>v v \<equiv> vec (dim_row A) (\<lambda>i. row A i \<bullet> v)"
bisim_simple_\<R>\<^sub>1: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val"   and \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and P :: "'BExp set" assumes "\<langle>c, mds, mem\<rangle> \<R>\<^sup>1\<^bsub>\<Gamma>,\<S>,P\<^esub> \<langle>c', mds', mem'\<rangle>" shows "c = c'"
condensation_test: fixes f :: "nat \<Rightarrow> real" assumes "\<And>m. 0 < m \<Longrightarrow> f (Suc m) \<le> f m"   and "\<And>n. 0 \<le> f n" shows "summable f = summable (\<lambda>n. 2 ^ n * f (2 ^ n))"
prems_8: shows "LeftDerivation [\<SS>] D (terminals (take r q) @ [N] @ \<gamma>)"
set_to_map_empty_iff_1: fixes S :: "('g \<times> 'h) set" shows "(set_to_map S = Map.empty) = (S = {})"
bernoulli_zero_iff: fixes n :: "nat" shows "(bernoulli n = 0) = (odd n \<and> n \<noteq> 1)"
GF_advice_inter_subformulas: fixes \<phi> :: "'b ltln"   and X :: "'b ltln set" shows "\<phi>[X \<inter> subformulas\<^sub>\<mu> \<phi>]\<^sub>\<nu> = \<phi>[X]\<^sub>\<nu>"
List_unique: fixes h :: "nat \<Rightarrow> nat option"   and x :: "nat"   and as :: "nat list"   and bs :: "nat list" assumes "List h x as"   and "List h x bs" shows "as = bs"
b: fixes x :: "'a" shows "(\<lambda>n. (F (n + 1) x - F n (T x)) * indicat_real A x) \<longlonglongrightarrow> f x * indicat_real A x"
simps_1: fixes s :: "'a FOL_Fitting.term"   and k :: "nat" shows "FF[s/k] = FF"
assms_1: fixes \<eta> :: "nat" shows "\<I>_res \<eta> \<turnstile>res res \<eta> \<surd>"
rtrancl_into_trancl2: fixes a :: "'c"   and b :: "'c"   and r :: "('c \<times> 'c) set"   and c :: "'c" assumes "(a, b) \<in> r"   and "(b, c) \<in> r\<^sup>*" shows "(a, c) \<in> r\<^sup>+"
assms_2: shows "\<exists>(e1, e2)\<in>edges G. \<not> p e1 \<le> p e2"
smult_assoc1: fixes a :: "'a"   and b :: "'a"   and x :: "'c" assumes "a \<in> carrier K"   and "b \<in> carrier K"   and "x \<in> carrier V" shows "a \<otimes>\<^bsub>K\<^esub> b \<odot>\<^bsub>V\<^esub> x = a \<odot>\<^bsub>V\<^esub> (b \<odot>\<^bsub>V\<^esub> x)"
simps_9: fixes \<E> :: "('a, 'b, 'c) cntxt"   and e :: "('a, 'b, 'c) expr" shows "(Rjoin\<^sub>\<E> \<E>) [e] = Rjoin (\<E> [e])"
simps_109: fixes x121 :: "expr"   and x122 :: "expr"   and x4 :: "val" shows "x121;; x122 \<noteq> Val x4"
pgalleryD_adj: fixes xs :: "'a set list" assumes "pgallery xs" shows "adjacentchain xs"
simps: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b stream" shows "holds P xs = P (shd xs)"
induct: fixes P :: "char list list \<Rightarrow> char list list \<Rightarrow> bool"   and a0 :: "char list list"   and a1 :: "char list list" assumes "\<And>p. P p []"   and "\<And>v va. P [] (v # va)"   and "\<And>x p y q. \<lbrakk>x = y \<Longrightarrow> P p q; \<lbrakk>x \<noteq> y; (x, y) \<in> var_order_rel\<rbrakk> \<Longrightarrow> P p (y # q); \<lbrakk>x \<noteq> y; (x, y) \<notin> var_order_rel\<rbrakk> \<Longrightarrow> P (x # p) q\<rbrakk> \<Longrightarrow> P (x # p) (y # q)" shows "P a0 a1"
component_constrains: fixes F :: "'d program"   and G :: "'d program"   and A :: "'d set"   and B :: "'d set" assumes "F \<le> G"   and "G \<in> A co B" shows "F \<in> A co B"
iso: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "isomorphic_categories A B" shows "\<exists>F G. inverse_functors A B F G"
Cons_1: fixes state :: "State"   and Q :: "Literal set"   and newWl :: "nat list" assumes "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "\<forall>c. c \<in> set Wl' \<longrightarrow> 0 \<le> c \<and> c < length (getF state)"   and "\<forall>c. c \<in> set Wl' \<longrightarrow> Some literal = getWatch1 state c \<or> Some literal = getWatch2 state c"   and "uniq Wl'"   and "getM state = M @ [(opposite literal, decision)]"   and "InvariantGetReasonIsReason (getReason state) (getF state) (getM state) Q" shows "let state' = notifyWatches_loop literal Wl' newWl state in Let (Q \<union> (set (getQ state') - set (getQ state))) (InvariantGetReasonIsReason (getReason state') (getF state') (getM state'))"
sum_fun_distl: fixes Y :: "'b set"   and x :: "'c"   and f :: "'b \<Rightarrow> 'c" assumes "finite Y" shows "x \<cdot> \<Sum> (f ` Y) = \<Sum> {x \<cdot> f y |y. y \<in> Y}"
simps: fixes pat :: "[:'r:]" shows "root2\<cdot>pat = rep2\<cdot>pat\<cdot>([::], pat)"
extend_fun: fixes f :: "'h \<Rightarrow> 'i"   and A :: "'h set"   and B :: "'i set"   and B1 :: "'i set" assumes "f \<in> A \<rightarrow> B"   and "B \<subseteq> B1" shows "f \<in> A \<rightarrow> B1"
infinite_Ioo: fixes a :: "'a"   and b :: "'a" assumes "a < b" shows "infinite {a<..<b}"
has_next_rbt_entries_generator: shows "list.has_next rbt_entries_generator = rbt_has_next"
is_full_pmdl_eq: fixes B1 :: "('t \<Rightarrow>\<^sub>0 'e) set"   and B2 :: "('t \<Rightarrow>\<^sub>0 'e) set" assumes "is_full_pmdl B1"   and "is_full_pmdl B2"   and "component_of_term ` Keys B1 = component_of_term ` Keys B2" shows "pmdl B1 = pmdl B2"
ValD_ValTT: fixes f :: "ValD \<rightarrow> ValD \<rightarrow> ValD" assumes "f\<cdot>\<bottom>\<cdot>ValTT = ValTT"   and "f\<cdot>ValTT\<cdot>\<bottom> = ValTT" shows "f\<cdot>ValTT\<cdot>ValTT = ValTT"
assms_2: shows "\<forall>l\<in>dom f. \<forall>x y z. Px x y z \<longrightarrow> P x y z (f l) (f' l)"
ext_inject: fixes hosts :: "'a set"   and flows_fix :: "('a \<times> 'a) set"   and flows_state :: "('a \<times> 'a) set"   and more :: "'b"   and hosts' :: "'a set"   and flows_fix' :: "('a \<times> 'a) set"   and flows_state' :: "('a \<times> 'a) set"   and more' :: "'b" shows "(\<lparr>hosts = hosts, flows_fix = flows_fix, flows_state = flows_state, \<dots> = more\<rparr> = \<lparr>hosts = hosts', flows_fix = flows_fix', flows_state = flows_state', \<dots> = more'\<rparr>) = (hosts = hosts' \<and> flows_fix = flows_fix' \<and> flows_state = flows_state' \<and> more = more')"
lb2_def: shows "lb2 = (\<lambda>n. (1 - 1 / ln (real n)) powr (ln (real n) / ln (ln (real n))))"
distinct_3: fixes x1 :: "'a" shows "mutator x1 \<noteq> sys"
and_mask_eq_iff_shiftr_0: fixes w :: "'c word"   and n :: "nat" shows "(w AND mask n = w) = (w >> n = 0)"
sorted_wrt_upt: fixes m :: "nat"   and n :: "nat" shows "Sorted_Less.sorted [m..<n]"
convex_with_affine_domain_is_lsc: fixes f :: "'b \<Rightarrow> ereal" assumes "Lower_Semicontinuous.convex_on UNIV f"   and "affine (domain f)" shows "lsc f"
esqrt_le: fixes x :: "ereal"   and y :: "ereal" shows "(esqrt x \<le> esqrt y) = (x \<le> 0 \<or> x \<le> y)"
processed_def: fixes p :: "'e \<times> 'e"   and xs :: "'e list"   and ps :: "('e \<times> 'e) list" shows "processed p xs ps = (p \<in> set xs \<times> set xs \<and> p \<notin>\<^sub>p set ps)"
character_principal: shows "character (G\<lparr>carrier := {\<one>}\<rparr>) (principal_char (G\<lparr>carrier := {\<one>}\<rparr>))"
Melem_subst_cls: fixes L :: "'a literal"   and C :: "'a literal multiset"   and \<sigma> :: "'s" shows "(L \<in># C \<cdot> \<sigma>) = (\<exists>M. M \<in># C \<and> L = M \<cdot>l \<sigma>)"
rep_eq: fixes x :: "'a"   and xa :: "'a poly" shows "coeff (smult x xa) = (\<lambda>n. x * coeff xa n)"
N': shows "N' = set (map (\<lambda>a. (post (delta a r'), post (delta a s'))) (\<sigma> n)) - N \<union> N"
distinct_delete_aux: fixes ps :: "('c \<times> 'd) list"   and k :: "'c" assumes "distinct (map fst ps)" shows "distinct (map fst (AList.delete_aux k ps))"
simps: fixes a1 :: "'varSort \<times> 'sort"   and a2 :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set" shows "wlsAbs a1 a2 = (\<exists>xs s X x. a1 = (xs, s) \<and> a2 = Abs xs x X \<and> isInBar (xs, s) \<and> wls s X)"
can_signal_trans: fixes s' :: "l1_state"   and A :: "agent"   and B :: "agent"   and s :: "l1_state" assumes "can_signal s' A B"   and "(s, s') \<in> TS.trans l1" shows "can_signal s A B"
a_A: fixes x :: "'a"   and y :: "'a" shows "ad (A x (ad y)) =  |x] y + ad y"
assms: shows "s \<rightarrow>\<^sub>\<beta> t"
simps: fixes s :: "nat"   and l :: "cell list"   and r :: "cell list"   and lm :: "nat list"   and n :: "nat"   and ires :: "cell list" shows "mopup_aft_erase_b (s, l, r) lm n ires = (\<exists>lnl lnr rn ml m. m = Suc (lm ! n) \<and> l = Bk \<up> lnr @ Oc \<up> m @ Bk \<up> lnl @ Bk # Bk # ires \<and> (r = Bk # <ml> @ Bk \<up> rn \<or> r = Bk # Bk # <ml> @ Bk \<up> rn))"
group_def: fixes G :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c" shows "Group_Theory.group G composition unit \<equiv> Group_Theory.monoid G composition unit \<and> Group_Theory.group_axioms G composition unit"
le_imp_big_d'_subset: fixes v :: "nat list"   and y :: "nat list"   and i :: "nat" assumes "v \<le>\<^sub>v y" shows "set (big_d' a b v i) \<subseteq> set (big_d' a b y i)"
indexed_var_not_index_free: fixes i :: "'c" shows "\<not> M.index_free (M.indexed_var i) i"
zero_comp_diff_simps_2: fixes a :: "'a"   and b :: "'a" shows "((0::'a) < a - b) = (b < a)"
emsapss_encode_help1_def: fixes mHash :: "bit list"   and emBits :: "nat" shows "emsapss_encode_help1 mHash emBits = (if emBits < length mHash + sLen + 16 then [] else emsapss_encode_help2 (generate_M' mHash salt) emBits)"
fps_coeff_residues_bound: fixes A :: "complex set"   and r :: "real"   and f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and k :: "complex set"   and C :: "real"   and n :: "nat" assumes "open A"   and "connected A"   and "cball 0 r \<subseteq> A"   and "0 < r"   and "f holomorphic_on A - S"   and "S \<subseteq> ball 0 r"   and "finite S"   and "0 \<notin> S"   and "\<And>z. \<lbrakk>cmod z = r; z \<notin> k\<rbrakk> \<Longrightarrow> cmod (f z) \<le> C"   and "0 \<le> C"   and "finite k" shows "cmod ((deriv ^^ n) f 0 / fact n + sum (residue (\<lambda>w. f w / w ^ Suc n)) S) \<le> C / r ^ n"
gen_check_filter: fixes as :: "nat list" shows "c2.gen_check as = filter (suffs C\<^sub>2 as) (alls B\<^sub>2 as)"
properL_def: fixes cl :: "('d, 'e, 'f) cmd list" shows "properL cl \<equiv> cl \<noteq> [] \<and> (\<forall>c\<in>set cl. proper c)"
not_updated_def: fixes r :: "nat"   and t :: "transition" shows "not_updated r t = (filter (\<lambda>(r', uu_). r' = r) (Updates t) = [])"
ar_coset_same1: fixes I :: "'a set"   and a :: "'a"   and b :: "'a" assumes "ideal R I"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "b \<plusminus> -\<^sub>a a \<in> I" shows "a \<uplus>\<^bsub>R\<^esub> I = b \<uplus>\<^bsub>R\<^esub> I"
integrable_Dirichlet_kernel: fixes n :: "nat" shows "integrable (lebesgue_on {- pi..pi}) (Dirichlet_kernel n)"
simps_8: fixes f :: "'e \<Rightarrow> 'd" shows "map_option f None = None"
span_eq: fixes S :: "'b set"   and T :: "'b set" assumes "S \<subseteq> U\<^sub>V\<^sub>S"   and "T \<subseteq> U\<^sub>V\<^sub>S" shows "(local.span S = local.span T) = (S \<subseteq> local.span T \<and> T \<subseteq> local.span S)"
finprod_insert: fixes F :: "'c set"   and a :: "'c"   and f :: "'c \<Rightarrow> 'a" assumes "finite F"   and "a \<notin> F"   and "f \<in> F \<rightarrow> carrier G"   and "f a \<in> carrier G" shows "finprod G f (insert a F) = f a \<otimes> finprod G f F"
row_add_def: fixes A :: "(('d, 'e) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec"   and a :: "'f"   and b :: "'f"   and q :: "'d" shows "row_add A a b q = (\<chi>i j. if i = a then A $h a $h j + q * A $h b $h j else A $h i $h j)"
pdouble_in_carrierp: fixes a :: "'a"   and p :: "'a \<times> 'a \<times> 'a" assumes "a \<in> carrier R"   and "in_carrierp p" shows "in_carrierp (pdouble a p)"
count_size_set_repr: fixes g :: "'b"   and A :: "'b multiset" shows "size {#x \<in># A. x = g#} = count A g"
exhaust_sel: fixes option :: "'k option" obtains "option = None"   | "option = Some (the option)"
bnd_1: fixes y :: "'b" assumes "y \<in> G" shows "norm y < bnd"
flows_reverse: fixes t :: "real"   and t0 :: "real"   and x0 :: "real" assumes "t \<in> existence_ivl t0 x0" shows "flow t (flow t0 x0 t) t0 = x0"
norm_axioms_def: fixes V :: "'b set"   and norm :: "'b \<Rightarrow> real" shows "norm_axioms V norm \<equiv> \<forall>x. x \<in> V \<longrightarrow> (norm x = 0) = (x = (0::'b))"
simps_2: fixes l :: "'c tree"   and uu :: "'c"   and r :: "'c tree" shows "min_height \<langle>l, uu, r\<rangle> = min (min_height l) (min_height r) + 1"
satC_e: 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 intT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool"   and \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and c :: "('a, 'b) lit list" assumes "M_FullModel wtFsym wtPsym arOf resOf parOf \<Phi> intT intF intP" shows "U.Struct.satC intF intP \<xi> c = M.Struct.satC intF intP \<xi> c"
outputCases': fixes a :: "name"   and b :: "name"   and P :: "pi"   and Rs :: "Late_Semantics.residual" assumes "a{b}.P \<longmapsto>\<^sub>l Rs" obtains aa :: "name" and ba :: "name" and Pa :: "pi" where "a = aa \<and> b = ba \<and> P = Pa" and "Rs = aa[ba] \<prec>\<^sub>l Pa"
less_le_not_le: fixes y :: "('d, 'e) Term.term"   and x :: "('d, 'e) Term.term" shows "y \<rhd> x = (y \<unrhd> x \<and> (x, y) \<notin> {\<unrhd>})"
POWSER_def: shows "POWSER \<equiv> \<lambda>f z. \<Sum>n. f n * z ^ n"
int_d2: shows "integrable (lebesgue_on {a..b}) (\<lambda>x. d\<^sup>2)"
case_distrib: fixes h :: "'c \<Rightarrow> 'b"   and f :: "'d match_expr \<Rightarrow> action \<Rightarrow> 'c"   and rule :: "'d rule" shows "h (case rule of Rule x xa \<Rightarrow> f x xa) = (case rule of Rule x1 x2 \<Rightarrow> h (f x1 x2))"
simps_2: fixes Q :: "('c \<Rightarrow> 'd option) \<Rightarrow> bool"   and cf :: "('c \<Rightarrow> 'd option) \<times> 'e"   and cfs :: "(('c \<Rightarrow> 'd option) \<times> 'e) list" shows "local.CFG.pred (Q)\<^sub>\<surd> (cf # cfs) = Q (fst cf)"
AxConeTangent: fixes onCone :: "'a Point \<Rightarrow> 'a Cone \<Rightarrow> bool"   and e :: "'a Point"   and cone :: "'a Cone"   and pt :: "'a Point" shows "onCone e cone \<longrightarrow> (inPlane pt (tangentPlane e cone) \<and> onCone pt cone) = collinear (vertex cone) e pt"
walk_edges_decomp': fixes xs :: "'a list"   and v :: "'a"   and w :: "'a"   and xs' :: "'a list" shows "edges_of_walk (xs @ v # w # xs') = edges_of_walk (xs @ [v]) \<union> {(v, w)} \<union> edges_of_walk (w # xs')"
image_mset_eq_imp_set_eq: fixes f :: "'c \<Rightarrow> 'b"   and s :: "'c multiset"   and g :: "'c \<Rightarrow> 'b" assumes "image_mset f s = image_mset g s" shows "f ` set_mset s = g ` set_mset s"
m2': shows "\<forall>x\<in>V \<times> V - E. \<not> all_security_requirements_fulfilled M \<lparr>nodes = V, edges = insert x E\<rparr>"
assms_5: shows "Suc (mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A)) \<noteq> nrows A"
cases: fixes a :: "('s \<times> ('c \<Rightarrow> real) set) pmf"   and st :: "'s \<times> ('c \<Rightarrow> real) set" assumes "a \<in> \<K> st" obtains l :: "'s" and R :: "('c \<Rightarrow> real) set" and R' :: "('c \<Rightarrow> real) set" where "a = return_pmf (l__, R')" and "st \<in> \<S>" and "st = (l__, R__)" and "R' \<in> Succ \<R> R__" and "R' \<subseteq> {v. v \<turnstile> PTA.inv_of A l__}"   | l :: "'s" and R :: "('c \<Rightarrow> real) set" and g :: "('c, real) cconstraint" and \<mu> :: "('c set \<times> 's) pmf" where "a = map_pmf (\<lambda>(X, l). (l, [X::=0]R)) \<mu>" and "st \<in> \<S>" and "st = (l, R)" and "(l, g, \<mu>) \<in> PTA.trans_of A" and "R \<subseteq> {v. v \<turnstile> g}"   | "a = return_pmf st"
C1: fixes y :: "name"   and R :: "ccs"   and T :: "ccs" assumes "y \<sharp> R" shows "(R \<parallel> \<lparr>\<nu>y\<rparr>T, \<lparr>\<nu>y\<rparr>(R \<parallel> T)) \<in> Rel"
1: shows "Ind (set [a]) (set b\<^sub>2)"
verit_comp_simplify_2: fixes a :: "'h" shows "a \<le> a"
Crypt_def: shows "Crypt \<equiv> (id ---> rep_msg ---> abs_msg) CRYPT"
infinite_arbitrarily_large: fixes A :: "'d set"   and n :: "nat" assumes "infinite A" shows "\<exists>B. finite B \<and> card B = n \<and> B \<subseteq> A"
intros_10: fixes P :: "pi" shows "P \<oplus> \<zero> \<equiv>\<^sub>e P"
sumr_diff_mult_const2: fixes f :: "nat \<Rightarrow> 'b"   and n :: "nat"   and r :: "'b" shows "sum f {..<n} - of_nat n * r = (\<Sum>i<n. f i - r)"
Sum: fixes z :: "ZF"   and x :: "ZF" shows "Elem z (HOLZF.Sum x) = (\<exists>y. Elem z y \<and> Elem y x)"