hcju/isabelleps huggingface dataset:
prems_2: shows "G \<subseteq>\<^sub>m execute_serial_plan_sas_plus I (op # \<psi>)"
Fvars_fm_simps_5: fixes A :: "fm"   and B :: "fm" shows "Fvars (A IMP B) = Fvars A \<union> Fvars B"
minus_equality: fixes y :: "'c"   and x :: "'c" assumes "y \<oplus>\<^bsub>M\<^esub> x = \<zero>\<^bsub>M\<^esub>"   and "x \<in> carrier M"   and "y \<in> carrier M" shows "\<ominus>\<^bsub>M\<^esub> x = y"
invD_1: shows "gs.lin_indpt_list (map of_int_hom.vec_hom fs)"
B: shows "B = [y \<leftrightarrow> x] \<cdot> A"
subsumes_def: fixes t :: "'a"   and s :: "'a" shows "(t <\<cdot> s) = (t \<le>\<cdot> s \<and> \<not> s \<le>\<cdot> t)"
assms_1: shows "\<exists>x. [A*\<^sub>vx]\<le>b"
uncountable_closed_segment: fixes a :: "'b"   and b :: "'b" assumes "a \<noteq> b" shows "uncountable (closed_segment a b)"
MkIde_Dom': fixes a :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr" assumes "J_C.ide a" shows "J_C.MkIde (J_C.Dom a) = a"
J_def: shows "J \<equiv> {i \<in> I. \<nexists>a. S i \<subseteq> {a}}"
d_equal_def: fixes DA :: "'c \<Rightarrow> 'd"   and d :: "'d"   and m :: "'c \<Rightarrow> 'e"   and m' :: "'c \<Rightarrow> 'e" shows "Strong_Security.d_equal DA d m m' \<equiv> \<forall>x. DA x \<le> d \<longrightarrow> m x = m' x"
assms: shows "(x2, y2) \<in> set (segments_of_aform X)"
acc_imp_SN_on: fixes x :: "'b"   and r :: "('b \<times> 'b) set" assumes "x \<in> Wellfounded.acc r" shows "SN_on {(y, z). (z, y) \<in> r} {x}"
new_subc_same'_1: fixes c :: "'a"   and s :: "'a FOL_Fitting.term"   and t :: "'a FOL_Fitting.term" assumes "new_term c s" shows "new_term c (subc_term c s t)"
gj_1: shows "(mat_of_rows n G)\<^sup>T \<in> carrier_mat n n"
allE': fixes P :: "'c \<Rightarrow> bool"   and x :: "'c" assumes "\<forall>x. P x" obtains "P x" and "\<forall>x. P x"
m_o_widen: fixes X :: "char list set"   and S1 :: "'av st option"   and S2 :: "'av st option" assumes "finite X"   and "top_on\<^sub>o S1 (- X)"   and "top_on\<^sub>o S2 (- X)"   and "\<not> S2 \<le> S1" shows "m\<^sub>o (S1 \<nabla> S2) X < m\<^sub>o S1 X"
lm080: fixes x :: "'c"   and f :: "('c \<times> 'd) set" assumes "x \<in> Domain f"   and "runiq f" shows "f = Outside' {x} f \<union> {(x, f ,, x)}"
node_ptr_no_document_ptr_cast: fixes document_ptr :: "(_) document_ptr" shows "\<not> is_node_ptr_kind (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document_ptr)"
iTILL_imirror_bounds: fixes n :: "nat"   and l :: "nat"   and r :: "nat" assumes "n \<le> l + r" shows "imirror_bounds [\<dots>n] l r = [l + r - n\<dots>,n]"
subdigraph_axioms: shows "\<AA> \<subseteq>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"
assms_6: fixes x :: "'c" assumes "x \<in> space T" shows "0 \<le> Py x"
sel_2: fixes x2 :: "'e" shows "projr (Inr x2) = x2"
last_ConsL: fixes xs :: "'b list"   and x :: "'b" assumes "xs = \<epsilon>" shows "last (x # xs) = x"
on_curve_infinity: fixes a :: "'a"   and b :: "'a" shows "on_curve a b Infinity"
interior_mono: fixes S :: "'c set"   and T :: "'c set" assumes "S \<subseteq> T" shows "interior S \<subseteq> interior T"
loc_lm_2: fixes n :: "nat" assumes "c_fst n mod 7 \<noteq> 0" shows "nat_to_sch n = (let u = c_fst n mod 7; v = c_snd n; v1 = c_fst v; v2 = c_snd v; sch1 = nat_to_sch v1 in Let (nat_to_sch v2) (loc_f u sch1))"
dim_direct_decomp: fixes A :: "('b \<Rightarrow>\<^sub>0 'c) set"   and ss :: "('b \<Rightarrow>\<^sub>0 'c) set list"   and B :: "('b \<Rightarrow>\<^sub>0 'c) set" assumes "direct_decomp A ss"   and "finite B"   and "A \<subseteq> phull B"   and "\<And>s. s \<in> set ss \<Longrightarrow> phull.subspace s" shows "phull.dim A = sum phull.dim (set ss)"
minus_1_div_2_eq: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and division_segment :: "'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c" assumes "class.unique_euclidean_ring_with_nat divide plus minus zero times one modulo euclidean_size division_segment uminus" shows "divide (uminus one) (numeral.numeral one plus (num.Bit0 num.One)) = uminus one"
ae: shows "AE x\<in>S in lebesgue. x \<in> {x \<in> closure S. f x = 0}"
llist_of_inject: fixes xs :: "'c list"   and ys :: "'c list" shows "(llist_of xs = llist_of ys) = (xs = ys)"
double_add_le_zero_iff_single_add_le_zero: fixes a :: "'b" shows "(a + a \<le> (0::'b)) = (a \<le> (0::'b))"
nb\<^sub>3: shows "dom ((\<phi>\<^sub>S \<Psi> s) ++ map_of (effect_to_assignments op')) = dom (\<phi>\<^sub>S \<Psi> s) \<union> set (effect_of op) \<union> (\<Union>(v, a)\<in>set (effect_of op). {(v, a') |a'. a' \<in> \<R>\<^sub>+ \<Psi> v \<and> a' \<noteq> a})"
Assms_3: shows "(s, s') \<in> obsv_round r S v D Ob"
permutation_mat_col: fixes p :: "nat \<Rightarrow> nat"   and n :: "nat"   and i :: "nat" assumes "p permutes {..<n}"   and "i < n" shows "col (permutation_mat n p) i = unit_vec n (p i)"
fun_upd_belowI: fixes x :: "'c"   and \<rho> :: "'c \<Rightarrow> 'd"   and \<rho>' :: "'c \<Rightarrow> 'd"   and y :: "'d" assumes "\<And>z. z \<noteq> x \<Longrightarrow> \<rho> z \<sqsubseteq> \<rho>' z"   and "y \<sqsubseteq> \<rho>' x" shows "\<rho>(x := y) \<sqsubseteq> \<rho>'"
prime_ideal_proper1: fixes p :: "'a set" assumes "prime_ideal R p" shows "1\<^sub>r \<notin> p"
assms_3: fixes x :: "'a" assumes "x \<in> S" shows "\<bar>f x\<bar> \<le> g x"
padic_add_res: fixes p :: "int"   and f :: "nat \<Rightarrow> int"   and g :: "nat \<Rightarrow> int"   and n :: "nat" shows "padic_add p f g n = f n \<oplus>\<^bsub>residue_ring (p ^ n)\<^esub> g n"
fps_zero_to_fls: shows "fps_to_fls 0 = 0"
adjoint_def: fixes A :: "'b mat" shows "adjoint A = mat (dim_col A) (dim_row A) (\<lambda>(i, j). conjugate (A $$ (j, i)))"
simps_2: fixes w :: "nat"   and t\<^sub>1 :: "'a tree"   and t\<^sub>2 :: "'a tree"   and w\<^sub>a :: "nat"   and a :: "'a"   and w\<^sub>b :: "nat"   and b :: "'a" shows "splitLeaf (Node w t\<^sub>1 t\<^sub>2) w\<^sub>a a w\<^sub>b b = Node w (splitLeaf t\<^sub>1 w\<^sub>a a w\<^sub>b b) (splitLeaf t\<^sub>2 w\<^sub>a a w\<^sub>b b)"
tan_bound_pi2: fixes x :: "real" assumes "\<bar>x\<bar> < pi / 4" shows "\<bar>tan x\<bar> < 1"
HT_partial_to_mod: fixes \<pi> :: "char list \<Rightarrow> com option"   and P :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool" shows "HT_partial \<pi> P c Q = HT_partial_mods \<pi> (ANALYZE (lhsv \<pi> c)) P c Q"
unfinished_dnodes_def: fixes w :: "'b \<times> 'b \<Rightarrow> enat"   and s :: "'b"   and S :: "'b set" shows "Dijkstra.unfinished_dnodes w s S \<equiv> Dijkstra.all_dnodes w s - S"
SatN: fixes A :: "'b set"   and H :: "(('c, 'b) fm list \<times> 'b) set"   and a :: "'b"   and i :: "'b"   and p :: "('c, 'b) fm" assumes "Hintikka A H"   and "(\<^bold>\<not> (\<^bold>@ i p)) at a in' H" shows "(\<^bold>\<not> p) at i in' H"
normalizePr'_def: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators" shows "normalizePr' p = (removeAllDuplicates \<circ> insertDenies \<circ> separate \<circ> sort' (Nets_List p) \<circ> removeShadowRules2 \<circ> remdups \<circ> rm_MT_rules Cp \<circ> insertDeny \<circ> removeShadowRules1 \<circ> policy2list) p"
simple_syntactic_or_axioms: shows "simple_syntactic_or shadows_consts"
suffix_tl: fixes xs :: "'b list" shows "suffix (tl xs) xs"
pz: assumes "w1 = POS_INF \<and> w2 = 0" shows "P w1 w2"
p: fixes k :: "char list"   and p :: "xml \<Rightarrow> ('a \<Rightarrow> 'b +\<^sub>\<bottom> 'c) \<Rightarrow> char list +\<^sub>\<bottom> 'd" assumes "(k, p) \<in> set ps" shows "monotone sum_bot.le_fun sum_bot_ord (p x)"
ldropn_Suc: fixes n :: "nat"   and xs :: "'c llist" shows "ldropn (Suc n) xs = (case xs of LNil \<Rightarrow> LNil | LCons x xs' \<Rightarrow> ldropn n xs')"
cases: fixes a1 :: "char list \<Rightarrow> secLevel option"   and a2 :: "expr"   and a3 :: "secLevel" assumes "a1 \<turnstile> a2 : a3" obtains \<Gamma> :: "char list \<Rightarrow> secLevel option" and V :: "val" and lev :: "secLevel" where "a1 = \<Gamma>'__" and "a2 = Val V" and "a3 = lev__"   | \<Gamma> :: "char list \<Rightarrow> secLevel option" and Vn :: "char list" and lev :: "secLevel" where "a1 = \<Gamma>''__" and "a2 = Var Vn" and "a3 = leva__" and "\<Gamma>''__ Vn = Some leva__"   | \<Gamma> :: "char list \<Rightarrow> secLevel option" and e1 :: "expr" and e2 :: "expr" and bop :: "bop" where "a1 = \<Gamma>'''__" and "a2 = e1__ \<guillemotleft>bop__\<guillemotright> e2__" and "a3 = Low" and "\<Gamma>'''__ \<turnstile> e1__ : Low" and "\<Gamma>'''__ \<turnstile> e2__ : Low"   | \<Gamma> :: "char list \<Rightarrow> secLevel option" and e1 :: "expr" and e2 :: "expr" and lev :: "secLevel" and bop :: "bop" where "a1 = \<Gamma>''''__" and "a2 = e1a__ \<guillemotleft>bopa__\<guillemotright> e2a__" and "a3 = High" and "\<Gamma>''''__ \<turnstile> e1a__ : High" and "\<Gamma>''''__ \<turnstile> e2a__ : levb__"   | \<Gamma> :: "char list \<Rightarrow> secLevel option" and e1 :: "expr" and lev :: "secLevel" and e2 :: "expr" and bop :: "bop" where "a1 = \<Gamma>" and "a2 = e1 \<guillemotleft>bop\<guillemotright> e2" and "a3 = High" and "\<Gamma> \<turnstile> e1 : lev" and "\<Gamma> \<turnstile> e2 : High"
epis_compose: fixes e :: "'a"   and e' :: "'a" assumes "epi e"   and "epi e'"   and "seq e' e" shows "epi (e' \<cdot> e)"
sem_env_def: fixes C_info :: "(name, dt_def) fmap"   and rs :: "(name \<times> sterm) list" shows "sem_env C_info rs = extend_dec_env (as_sem_env C_info rs (Compiler.empty_sem_env C_info)) (Compiler.empty_sem_env C_info)"
le_iff_sup: fixes x :: "'a"   and y :: "'a" shows "(x \<le> y) = (x + y = y)"
poly_to_mpoly0: fixes v :: "nat" shows "poly_to_mpoly v 0 = 0"
prefix_replicate_zero_coeff: fixes p :: "'a list"   and n :: "nat" shows "local.coeff p = local.coeff (replicate n \<zero> @ p)"
InfModel: shows "IInfModel I_intT I_intF I_intP"
job_dist_lower_bound_makespan: fixes T :: "nat \<Rightarrow> nat"   and A :: "nat \<Rightarrow> nat set"   and j :: "nat" assumes "lb T A j" shows "real (sum t {1..j}) / real m \<le> real (makespan T)"
next_def: fixes s :: "('c, 'd) configuration stream" shows "next s = (next_performop s \<or> next_sendupd s \<or> next_recvupd s \<or> shd (stl s) = shd s)"
rep_eq: fixes x :: "'a upred"   and xa :: "'a upred" shows "\<lbrakk>x \<Leftrightarrow> xa\<rbrakk>\<^sub>e = (\<lambda>A. \<lbrakk>x\<rbrakk>\<^sub>e A = \<lbrakk>xa\<rbrakk>\<^sub>e A)"
fold_random_permutation_foldr: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c" assumes "finite A" shows "fold_random_permutation f x A = map_pmf (\<lambda>xs. foldr f xs x) (pmf_of_set (permutations_of_set A))"
A2: shows "A2 = And (reset' A1 n r v (0::'c)) D'"
P5: shows "Bet B' A' A1 \<and> Cong A' A1 B'' A''"
rel_mono: fixes R1 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R1a :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R2 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and R2a :: "'f \<Rightarrow> 'g \<Rightarrow> bool" assumes "R1 \<le> R1a"   and "R2 \<le> R2a" shows "rel_nondetT' R1 R2 \<le> rel_nondetT' R1a R2a"
vsv_def: fixes r :: "V" shows "vsv r \<equiv> vbrelation r \<and> vsv_axioms r"
ps_not_Nil: shows "sturm_squarefree p \<noteq> []"
mem_upI: fixes f :: "nat \<Rightarrow> 'c"   and R :: "('c, 'd) ring_scheme" assumes "\<And>n. f n \<in> carrier R"   and "\<exists>n. bound \<zero>\<^bsub>R\<^esub> n f" shows "f \<in> up R"
eq_key_imp_eq_value: fixes xs :: "('a \<times> 'b) list"   and k :: "'a"   and v1 :: "'b"   and v2 :: "'b" assumes "distinct (map fst xs)"   and "(k, v1) \<in> set xs"   and "(k, v2) \<in> set xs" shows "v1 = v2"
mult_lc_monic: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "p \<noteq> 0" shows "monom_mult (lc p) (0::'a) (monic p) = p"
path_independent_def: fixes f :: "'d set \<Rightarrow> 'd set" shows "path_independent f = (\<forall>B C. f (B \<union> C) = f (B \<union> f C))"
parTrace_def: shows "parTrace \<equiv> rec_list False (\<lambda>cf cfl cfla. cfl \<noteq> [] \<longrightarrow> cfla \<and> cf \<rightarrow>c hd cfl)"
isCont_inverse_function2: fixes a :: "real"   and x :: "real"   and b :: "real"   and g :: "real \<Rightarrow> real"   and f :: "real \<Rightarrow> real" assumes "a < x"   and "x < b"   and "\<And>z. \<lbrakk>a \<le> z; z \<le> b\<rbrakk> \<Longrightarrow> g (f z) = z"   and "\<And>z. \<lbrakk>a \<le> z; z \<le> b\<rbrakk> \<Longrightarrow> isCont f z" shows "isCont g (f x)"
mat_mult_left_right_inverse: fixes A :: "'a mat"   and n :: "nat"   and B :: "'a mat" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n"   and "A * B = 1\<^sub>m n" shows "B * A = 1\<^sub>m n"
cake_times_def: fixes t :: "exp0"   and u :: "exp0" shows "cake_times t u = App (Opn Times) [t, u]"
mset_zero_iff: fixes x :: "'a list" shows "(mset x = {#}) = (x = [])"
ltl_prop_implies_equiv: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "\<phi> \<sim>\<^sub>P \<psi> = (\<phi> \<longrightarrow>\<^sub>P \<psi> \<and> \<psi> \<longrightarrow>\<^sub>P \<phi>)"
map_sel_2: fixes f :: "'d \<Rightarrow> 'c"   and a :: "'d sctn" shows "pstn (map_sctn f a) = pstn a"
fresh_Iff: fixes a :: "atom"   and A :: "fm"   and B :: "fm" shows "a \<sharp> A IFF B = (a \<sharp> A \<and> a \<sharp> B)"
f_def: shows "f = (\<lambda>n a i j k. norm ((jordan_block n a ^\<^sub>m k) $$ (i, j)))"
less_eq_acom_annos: fixes C1 :: "'a acom"   and C2 :: "'a acom" shows "(C1 \<le> C2) = (strip C1 = strip C2 \<and> list_all2 (\<le>) (annos C1) (annos C2))"
Sum_any_def: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c" assumes "class.comm_monoid_add plus zero" shows "comm_monoid_add.Sum_any plus zero \<equiv> comm_monoid_fun.G plus zero"
insert_mono: fixes C :: "'b set"   and D :: "'b set"   and a :: "'b" assumes "C \<subseteq> D" shows "a \<triangleright> C \<subseteq> a \<triangleright> D"
Cons_3: shows "length m2 = nc \<and> (\<forall>a\<in>set m2. vec nr a)"
netgmap_pair_dom: fixes sr :: "'f \<Rightarrow> 'd \<times> 'e"   and n :: "'f net_state" obtains \<sigma> :: "nat \<Rightarrow> 'd option" and \<zeta> :: "'e net_state" where "netgmap sr n = (\<sigma>, \<zeta>)" and "dom \<sigma> = net_ips n"
assms_3: shows "(n', nid') \<in> his t n nid"
tab_in_vhom': shows "\<guillemotleft>\<rho>' : g' \<Rightarrow> r \<star> f'\<guillemotright>"
inf_idem: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "class.semilattice_inf inf less_eq less" shows "inf x x = x"
minus_eq: fixes x :: "'b"   and y :: "'b" shows "x - y = x + - y"
stk_strict_subs_def: fixes S :: "'b set" shows "stk_strict_subs S = (S \<noteq> UNIV)"
1: fixes ars :: "String.literal \<Rightarrow> String.literal set list option"   and c\<^sub>1 :: "String.literal"   and ss1 :: "String.literal set list"   and c\<^sub>2 :: "String.literal"   and ss2 :: "String.literal set list" assumes "ars \<in> ran (map_of (map (apsnd map_of) arss))"   and "ars c\<^sub>1 = Some ss1"   and "ars c\<^sub>2 = Some ss2"   and "class_leq (set cs) c\<^sub>1 c\<^sub>2" shows "list_all2 (sort_leq (set cs)) ss1 ss2"
assms_1: fixes x :: "'a" shows "(\<lambda>y. f (x, y)) abs_summable_on Y"
gen_qe_eval_augment: fixes amount :: "nat"   and var :: "nat"   and L :: "atom list"   and F :: "atom fm list"   and \<Gamma> :: "real list" assumes "amount \<le> var + 1" shows "(\<exists>xs. length xs = var + 1 \<and> eval (list_conj (map fm.Atom L @ F)) (xs @ \<Gamma>)) = (\<exists>xs. length xs = var + 1 \<and> eval (step_augment gen_qe IdentityHeuristic amount var L F) (xs @ \<Gamma>))"
ij_le2_2: shows "j < prod_list (nths (dims A) (- I))"
power_overflow: fixes n :: "nat" assumes "LENGTH('b) \<le> n" shows "2 ^ n = 0"
has_derivative_proj: fixes f :: "'d \<Rightarrow> real \<Rightarrow> real"   and f' :: "'d \<Rightarrow> real \<Rightarrow> real"   and F :: "real filter"   and j :: "'d" assumes "((\<lambda>x. \<chi>i. f i x) has_derivative (\<lambda>h. \<chi>i. f' i h)) F" shows "(f j has_derivative f' j) F"
propagate_CLAMP_5: fixes a :: "real" shows "CLAMP (\<lambda>x. real x powr a) = CLAMP' (MASTER_BOUND a 0 0)"
game_B_def: shows "game_B = exec_gpv prf.random_oracle \<A> Map.empty \<bind> (\<lambda>(b, \<sigma>). local.hash.game_hash_set (dom \<sigma>))"
terminal_def: fixes b :: "'a \<times> 'a" shows "CC.terminal b \<equiv> CC.ide b \<and> (\<forall>a. CC.ide a \<longrightarrow> (\<exists>!f. CC.in_hom f a b))"
has_fps_expansion_mult: fixes f :: "'a \<Rightarrow> 'a"   and F :: "'a fps"   and g :: "'a \<Rightarrow> 'a"   and G :: "'a fps" assumes "f has_fps_expansion F"   and "g has_fps_expansion G" shows "(\<lambda>x. f x * g x) has_fps_expansion F * G"
assms_2: shows "q \<in> sterms \<Gamma> p"
order_greater_eq_two_iff: fixes f :: "'b perm"   and a :: "'b" shows "(2 \<le> order f a) = (a \<in> affected f)"
pinv: shows "A \<TTurnstile> (I \<rightarrow>) P"
weak_complete_lattice_def: fixes L :: "('a, 'b) gorder_scheme" shows "weak_complete_lattice L \<equiv> weak_partial_order L \<and> weak_complete_lattice_axioms L"
single_valued_as_brE: fixes R :: "('c \<times> 'd) set" assumes "single_valued R" obtains \<alpha> :: "'c \<Rightarrow> 'd" and invar :: "'c \<Rightarrow> bool" where "R = br \<alpha> invar"
segment_inc: fixes D :: "('c, 'd) Order_scheme"   and a :: "'c"   and b :: "'c" assumes "Order D"   and "a \<in> carrier D"   and "b \<in> carrier D" shows "a \<prec>\<^bsub>D\<^esub> b = (a \<in> segment D b)"
assms_1: shows "akra_bazzi_term x\<^sub>0 (Suc x\<^sub>1) b t"
F2set2_Lev2: fixes xf :: "(bd_type_F + bd_type_F) list"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('g, 'd, 'e) F2"   and n :: "nat"   and xd :: "'e"   and xb :: "'e"   and xa :: "'d" assumes "xf \<in> Lev2 s1 s2 n xd"   and "rv2 s1 s2 xf xd = Inr xb"   and "xa \<in> F2set2 (s2 xb)" shows "xf @ [Inl (tobd_F22 s2 xb xa)] \<in> Lev2 s1 s2 (Suc n) xd"
two_phase_def: fixes r :: "nat" shows "two_phase r \<equiv> r div 2"
add_closed_converse_left: fixes g :: "'g"   and x :: "'g" assumes "g + x \<in> H"   and "x \<in> H" shows "g \<in> H"
const_poly_dvd_iff: fixes c :: "'a"   and p :: "'a poly" shows "([:c:] dvd p) = (\<forall>n. c dvd coeff p n)"
xs_def: shows "xs = coeffs p @ replicate (n - length (coeffs p)) (0::'a)"
assms_2: shows "\<forall>n'>n. \<not> P n'"
prems: shows "App e\<^sub>1 e\<^sub>2 \<Down> t\<^sub>2"
simps_1: fixes u :: "'a" shows "pawalk_verts u [] = [u]"
ennreal_mult_less_top: fixes a :: "ennreal"   and b :: "ennreal" shows "(a * b < \<top>) = (a = 0 \<or> b = 0 \<or> a < \<top> \<and> b < \<top>)"
assms_2: shows "\<forall>A\<in>S. ideal A"
unifiers_insert: fixes p :: "('d, 'c) Term.term \<times> ('d, 'c) Term.term"   and t :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) set" shows "unifiers (insert p t) = {\<sigma>. fst p \<cdot> \<sigma> = snd p \<cdot> \<sigma>} \<inter> unifiers t"
vpeq_local_def: fixes u :: "partition_id_t"   and s :: "state_t"   and t :: "state_t" shows "vpeq_local u s t \<equiv> \<forall>tid. Step.partition tid = u \<longrightarrow> thread s tid = thread t tid"
is_processT5_S4: fixes s :: "'a event list"   and P :: "'a process"   and c :: "'a event" assumes "(s, {}) \<in> F P"   and "(s, {c}) \<notin> F P" shows "(s @ [c], {}) \<in> F P"
cases: fixes h :: "'c \<Rightarrow> 'd"   and bo :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a1 :: "('c, 'd) blindable\<^sub>m"   and a2 :: "('c, 'd) blindable\<^sub>m" assumes "blinding_of_blindable h bo a1 a2" obtains x :: "'c" and y :: "'c" where "a1 = Unblinded xa__" and "a2 = Unblinded y" and "bo xa__ y"   | t :: "('c, 'd) blindable\<^sub>m" and x :: "'d blindable\<^sub>h" where "a1 = Blinded x" and "a2 = t" and "hash_blindable h t = x"
finite: shows "finite {p. prime p \<and> real p \<le> x}"
prime_elem_dvd_power_iff: fixes p :: "'a"   and a :: "'a"   and n :: "nat" assumes "prime_elem p" shows "(p dvd a ^ n) = (p dvd a \<and> 0 < n)"
roots_def: fixes p :: "real poly" shows "roots p = {x. poly p x = 0}"
S22: shows "pr_gr upb = c_cons (c_pair (c_pair n x) (c_pair (c_assoc_value (pr_gr (c_pair (c_pair n x) (upb1 + upb2))) (c_pair (c_fst (c_snd n)) x)) (c_assoc_value (pr_gr (c_pair (c_pair n x) (upb1 + upb2))) (c_pair (c_snd (c_snd n)) x)))) (pr_gr (c_pair (c_pair n x) (upb1 + upb2)))"
simps_9: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltl" shows "w \<Turnstile> F \<phi> = (\<exists>k. suffix k w \<Turnstile> \<phi>)"
less_rexp_def: fixes r :: "'b rexp"   and s :: "'b rexp" shows "r < s \<equiv> le_rexp r s \<and> r \<noteq> s"
assms: shows "fbrelation (r \<union>\<^sub>\<circ> s)"
Exit_path: fixes n :: "'node" assumes "valid_node n" shows "\<exists>as. n -as\<rightarrow>* (_Exit_)"
Int_atLeastAtMostL1: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and d :: "'a" assumes "class.linorder less_eq less" shows "ord.atLeastAtMost less_eq a b \<inter> ord.atMost less_eq d = ord.atLeastAtMost less_eq a (ord.min less_eq b d)"
sim: shows "weak_reduction_simulation (Rel\<inverse>) (STCal Source Target)"
assms_3: shows "(l', R') \<in> set_pmf \<mu>"
Dfb_def: fixes x :: "'a" shows "Dfb x = Blinfun (Df x)"
closed_csubspace_def: fixes A :: "'a set" shows "closed_csubspace A \<equiv> csubspace A \<and> closed A"
unit_char_2: fixes a :: "'a" assumes "obj a" shows "S.iso (UP.unit a)"
seq_Suc: fixes i :: "nat" shows "seq (Suc i) = fun (seq i)"
landau_product_preprocess_14: fixes n :: "num" shows "num.One * n = n"
inject_2: fixes x2 :: "ty"   and x1 :: "ty"   and y2 :: "ty"   and y1 :: "ty" shows "(x2 \<rightarrow> x1 = y2 \<rightarrow> y1) = (x2 = y2 \<and> x1 = y1)"
B_trusts_NS3: fixes B :: "agent"   and K :: "nat"   and A :: "agent"   and evs :: "event list" assumes "Crypt (shrK B) \<lbrace>Key K, Agent A\<rbrace> \<in> parts (knows Spy evs)"   and "B \<notin> bad"   and "evs \<in> ns_shared" shows "\<exists>NA. Says Server A (Crypt (shrK A) \<lbrace>NA, Agent B, Key K, Crypt (shrK B) \<lbrace>Key K, Agent A\<rbrace>\<rbrace>) \<in> set evs"
o1: shows "m = 3 ^ 3 * 5 * 11 * 17 * 227"
basis_def: fixes A :: "'a Matrix.vec set" shows "basis A = (lin_indpt A \<and> local.span A = mat_kernel (char_matrix A lam) \<and> A \<subseteq> mat_kernel (char_matrix A lam))"
harm_asymp_equiv: shows "sum_upto (\<lambda>n. 1 / real n) \<sim>[at_top] ln"
inter_union_distrib_left: fixes A :: "'k multiset"   and B :: "'k multiset"   and C :: "'k multiset" shows "A \<inter># B + C = (A + C) \<inter># (B + C)"
dest_in_sdnets: fixes x :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and p :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent" assumes "\<not> member DenyAll x"   and "p \<in> dom (Cp x)" shows "subnetsOfAdr (dest p) \<inter> snd_set (sdnets x) \<noteq> {}"
countable_reachable: fixes s :: "'a cfg" shows "countable (MDP.MC.acc `` {s})"
case_two_nontotal_hyp: fixes j :: "nat" assumes "\<exists>t. local.prefixes t j \<up>" shows "\<exists>n<longest_prefix j. \<not> total1 (\<phi> (the (s (\<psi> j \<triangleright> n))))"
more_arith_simps_11: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a \<cdot> b \<cdot> c = a \<cdot> (b \<cdot> c)"
int_simps_15: fixes P :: "'a \<Rightarrow> bool" shows "\<turnstile> (P \<longrightarrow> \<not> P) = \<not> P"
SUP_cong_simp: fixes A :: "'e set"   and B :: "'e set"   and C :: "'e \<Rightarrow> 'f"   and D :: "'e \<Rightarrow> 'f"   and Sup :: "'f set \<Rightarrow> 'f" assumes "A = B"   and "\<And>x. x \<in> B =simp=> C x = D x" shows "Sup (C ` A) = Sup (D ` B)"