hcju/isabelleps huggingface dataset:
order_divides: fixes a :: "'c"   and n :: "nat"   and p :: "'c poly" shows "([:- a, 1::'c:] ^ n dvd p) = (p = 0 \<or> n \<le> order a p)"
is_free_simps_2: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and name :: "name" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "\<not> pre_term.is_free unfree (const name)"
positive_def: fixes M :: "'d set set"   and \<mu> :: "'d set \<Rightarrow> ennreal" shows "positive M \<mu> = (\<mu> {} = 0)"
that: fixes N :: "nat set"   and js :: "nat list" assumes "infinite N"   and "\<And>k u. \<lbrakk>k < 4; u \<in> [UU]\<^bsup>2\<^esup>; form u k; scheme u \<subseteq> N\<rbrakk> \<Longrightarrow> f u = js ! k" shows "thesis"
indexlt_simp: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and k' :: "nat"   and u' :: "nat"   and k :: "nat"   and u :: "nat" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel" shows "indexlt k' u' k u = (k' < k \<or> k' = k \<and> u' < u)"
vid_on_vimage: fixes A :: "V"   and B :: "V" shows "vid_on A `\<^sub>\<circ> B = A \<inter>\<^sub>\<circ> B"
assms_2: shows "cpt (P - {p}) env1 \<pi>l1"
simps: fixes \<alpha>1 :: "'b"   and P1 :: "'c"   and \<alpha>2 :: "'b"   and P2 :: "'c" shows "alpha_residual (\<alpha>1, P1) (\<alpha>2, P2) = ([bn \<alpha>1]set. (\<alpha>1, P1) = [bn \<alpha>2]set. (\<alpha>2, P2))"
HEndPhase0_Inv6: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HEndPhase0 s s' p"   and "HInv6 s"   and "Inv1 s"   and "Inv2c s" shows "HInv6 s'"
full_sum: fixes S :: "'a set"   and p :: "real" assumes "finite S" shows "(\<Sum>A\<in>Pow S. p ^ card A * (1 - p) ^ card (S - A)) = 1"
limit_cycle_def: fixes y :: "'a" shows "limit_cycle y = (periodic_orbit y \<and> (\<exists>x. x \<notin> range (flow0 y) \<and> (range (flow0 y) = \<omega>_limit_set x \<or> range (flow0 y) = \<alpha>_limit_set x)))"
distinct_sorted_list_of_set: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "class.linorder less_eq less" shows "distinct (linorder.sorted_list_of_set less_eq A)"
squarefree_mult_coprime: fixes a :: "'b"   and b :: "'b" assumes "coprime a b"   and "squarefree a"   and "squarefree b" shows "squarefree (a * b)"
in_ivl_affine_of_ivlE: fixes k :: "'a"   and l :: "'a"   and u :: "'a" assumes "k \<in> {l..u}" obtains e :: "nat \<Rightarrow> real" where "e \<in> UNIV \<rightarrow> {- 1..1}" and "k = aform_val e (aform_of_ivl l u)"
cases: fixes x :: "nat \<times> (bool list \<times> bool list) insec_query" obtains \<eta> :: "nat" and a :: "bool list" and m :: "bool list" where "x = (\<eta>__, Edit (a, m))"   | \<eta> :: "nat" where "x = (\<eta>'__, Look)"   | \<eta> :: "nat" where "x = (\<eta>''__, Forward)"   | \<eta> :: "nat" where "x = (\<eta>, Drop)"
all_dbm: shows "\<exists>M. (\<Union> \<R> = [M]\<^bsub>v,n\<^esub> \<and> dbm_int M n) \<and> normalized M"
isCont_powser: fixes c :: "nat \<Rightarrow> 'a"   and K :: "'a"   and x :: "'a" assumes "summable (\<lambda>n. c n * K ^ n)"   and "norm x < norm K" shows "isCont (\<lambda>x. \<Sum>n. c n * x ^ n) x"
simps_1: fixes s :: "'c stream" shows "sdrop 0 s = s"
fine_Union: fixes ps :: "('a \<times> 'b set) set set"   and d :: "'a \<Rightarrow> 'b set" assumes "\<And>p. p \<in> ps \<Longrightarrow> d fine p" shows "d fine \<Union> ps"
preserves_hom: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>A b\<guillemotright>" shows "\<guillemotleft>\<phi> f : F a \<rightarrow>\<^sub>B G b\<guillemotright>"
c_len_def: shows "c_len = (\<lambda>u. sgn1 u * (c_fst (u - 1) + 1))"
a0i_defs_7: shows "a0i \<equiv> \<lparr>init = a0i_init, trans = a0i_trans, obs = id\<rparr>"
\<epsilon>_is_natural_transformation: shows "natural_transformation (\<cdot>\<^sub>A) (\<cdot>\<^sub>A) FG.map A.map \<epsilon>"
initial_state'_def: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list" shows "initial_state' problem \<equiv> let astI = ast_problem.astI problem in (\<lambda>v. if v < length astI then Some (astI ! v) else None)"
div_unit_factor: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "divide a (unit_factor a) = normalize a"
eq: shows "standard_simplex p = (\<Inter>i. {x \<in> topspace (powertop_real UNIV). x i \<in> {0..1}}) \<inter> (\<Inter>i\<in>{p<..}. {x \<in> topspace (powertop_real UNIV). x i \<in> {0}}) \<inter> {x \<in> topspace (powertop_real UNIV). sum x {..p} \<in> {1}}"
zero_iff: fixes x :: "'a" assumes "x \<in> V" shows "(\<parallel>x\<parallel> = 0) = (x = (0::'a))"
balanced_pos_and_neg_terms_cancel_out1: assumes "is_balanced" shows "(\<Sum>k<2 ^ n. (- int 1) ^ f k) = 0"
intro: fixes \<CC> :: "V"   and \<BB> :: "V" assumes "\<And>a. a \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr> \<Longrightarrow> a \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>" shows "wide_subdigraph_axioms \<BB> \<CC>"
wordinterval_compress_def: fixes r :: "'b wordinterval" shows "wordinterval_compress r \<equiv> l2wi (remdups (listwordinterval_adjacent (listwordinterval_compress (wi2l (wordinterval_optimize_empty r)))))"
fold_congs_1: fixes r :: "'a point_scheme"   and r' :: "'a point_scheme"   and v' :: "nat"   and f :: "nat \<Rightarrow> nat"   and f' :: "nat \<Rightarrow> nat" assumes "r = r'"   and "xpos r' = v'"   and "\<And>v. v' = v \<Longrightarrow> f v = f' v" shows "xpos_update f r = xpos_update f' r'"
residue_primroot_1_iff: fixes n :: "nat" shows "residue_primroot n (Suc 0) = (n \<in> {1, 2})"
indRelRSTPO_impl_SRel_and_TRel_reflect_barbs_1: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "rel_reflects_binary_pred (indRelRSTPO SRel TRel) (\<lambda>b. hasBarb b (STCalWB SWB TWB))" shows "rel_reflects_binary_pred SRel (\<lambda>b. hasBarb b SWB)"
lmap_llist_of_stream: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d stream" shows "lmap f (llist_of_stream xs) = llist_of_stream (smap f xs)"
id_cblinfun_adjoint: shows "id_cblinfun* = id_cblinfun"
mono_update: fixes a1 :: "'b"   and a2 :: "'b"   and S1 :: "'b st"   and S2 :: "'b st"   and x :: "char list" assumes "a1 \<le> a2"   and "S1 \<le> S2" shows "update S1 x a1 \<le> update S2 x a2"
dg_Rel_Hom_vifunion_in_Vset: fixes X :: "V"   and Y :: "V" assumes "X \<in>\<^sub>\<circ> Vset \<alpha>"   and "Y \<in>\<^sub>\<circ> Vset \<alpha>" shows "\<Union>\<^sub>\<circ> ((\<lambda>a\<in>\<^sub>\<circ>X. \<Union>\<^sub>\<circ> (VLambda Y (Hom (dg_Rel \<alpha>) a) `\<^sub>\<circ> Y)) `\<^sub>\<circ> X) \<in>\<^sub>\<circ> Vset \<alpha>"
left_neutral: fixes a :: "'a" shows "\<zero> \<^bold>\<squnion> a = a"
assms: fixes e :: "'a uprod" assumes "e \<in> F" shows "proper_uprod e"
\<rho>_agreement: shows "\<rho> = (\<lambda>f. if arr f then f \<cdot> \<r>[local.dom f] else null)"
section_retraction_of_iso_1: fixes f :: "'b"   and g :: "'b" assumes "B.iso f"   and "B.ide (g \<cdot>\<^sub>B f)" shows "B.inverse_arrows f g"
cccFix_def: fixes cccExp :: "exp \<Rightarrow> Arity \<rightarrow> (Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> CoCalls"   and \<Gamma> :: "(Vars.var \<times> exp) list" shows "CoCallArityAnalysis.cccFix cccExp \<Gamma> = (\<Lambda> i. (ArityAnalysis.Afix (CoCallArityAnalysis.Aexp cccExp) \<Gamma>\<cdot>(fst i \<squnion> (\<lambda>_. up\<cdot>0) f|` thunks \<Gamma>), CoCallArityAnalysis.CCfix cccExp \<Gamma>\<cdot>(ArityAnalysis.Afix (CoCallArityAnalysis.Aexp cccExp) \<Gamma>\<cdot>(fst i \<squnion> (\<lambda>_. up\<cdot>0) f|` thunks \<Gamma>), snd i)))"
bitlen_ge_iff_power: fixes w :: "int"   and x :: "int" shows "(w \<le> bitlen x) = (w \<le> 0 \<or> 2 ^ (nat w - 1) \<le> x)"
simps_2: fixes uu :: "('b \<times> nat) tree"   and uv :: "'b"   and n :: "nat"   and uw :: "('b \<times> nat) tree" shows "size_wbt \<langle>uu, (uv, n), uw\<rangle> = n"
bidi_4: fixes a :: "'b" assumes "a \<in> arcs G" shows "tail G (edge_rev M a) = head G a"
semiring_simprules_6: fixes R :: "('b, 'c) ring_scheme"   and x :: "'b" assumes "semiring R"   and "x \<in> carrier R" shows "\<zero>\<^bsub>R\<^esub> \<oplus>\<^bsub>R\<^esub> x = x"
snd_apsnd: fixes f :: "'f \<Rightarrow> 'd"   and x :: "'e \<times> 'f" shows "snd (apsnd f x) = f (snd x)"
rga_spec_equal: fixes xs :: "('b \<times> 'b option) list"   and ys :: "('b \<times> 'b option) list" assumes "set xs = set ys"   and "insert_ops xs"   and "rga_ops ys" shows "interp_ins xs = interp_rga ys"
ln_upper_15_def: fixes x :: "real" shows "ln_upper_15 x \<equiv> (1487 + 49199 * x + 547235 * x ^ 4 + 718735 * x ^ 3 + 334575 * x\<^sup>2 + 141123 * x ^ 5 + 35 * x ^ 7 + 9411 * x ^ 6) * (x - 1) / (280 * (1 + 56 * x + 2450 * x ^ 4 + 1960 * x ^ 3 + 588 * x\<^sup>2 + 1176 * x ^ 5 + 8 * x ^ 7 + 196 * x ^ 6))"
bind_pmf_spmf_assoc: fixes g :: "'d option \<Rightarrow> 'c spmf"   and p :: "'e spmf"   and f :: "'e \<Rightarrow> 'd spmf" assumes "g None = return_pmf None" shows "p \<bind> f \<bind> g = p \<bind> (\<lambda>x. f x \<bind> g)"
tbe: shows "tighten_poly_bounds_epsilon p (1 / 2) l r (sgn (ipoly p r)) = (l', r', sr')"
exp1': fixes A :: "nat"   and B :: "nat"   and C :: "nat" shows "exp 3 A < 3 * (exp 3 A * exp 3 B) + C"
matrix_inv_right: fixes M :: "(('b, 'c) vec, 'd) vec" assumes "invertible M" shows "M ** matrix_inv M = mat (1::'b)"
IH: fixes ys :: "'a list" assumes "strict_sorted xs"   and "strict_sorted ys"   and "set ys = set xs" shows "ys = xs"
ereal_zero_times: fixes a :: "ereal"   and b :: "ereal" shows "(a * b = 0) = (a = 0 \<or> b = 0)"
1: shows "top = (THE x. \<forall>y. x = y \<squnion> - y)"
inf_ordering_img: fixes X :: "'b set"   and f :: "nat \<Rightarrow> 'b"   and ord :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "infinite X"   and "TernaryOrdering.ordering f ord X" shows "f ` {0..} = X"
is_map: shows "is_left_adjoint Tfg_Hfg.chine"
intro: fixes S :: "'c set"   and T :: "'d set" assumes "real_vector_space_on S"   and "real_vector_space_on T" shows "real_vector_space_pair_on S T"
foldl_assoc: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and xs :: "'a list" assumes "\<And>x y z. f (f x y) z = f x (f y z)" shows "foldl f (f a b) xs = f a (foldl f b xs)"
create_opt_sol_satisfies_geq0: fixes A :: "rat mat"   and b :: "rat vec"   and c :: "rat vec"   and X :: "(nat, rat) mapping"   and x :: "rat vec"   and y :: "rat vec"   and i :: "nat" assumes "create_optimal_solutions A b c = Inr X"   and "(x, y) = split_n_m_x (dim_vec c) (dim_vec b) X"   and "i < dim_vec y" shows "0 \<le> y $ i"
Sup_pres_Sup_closed: fixes f :: "'b \<Rightarrow> 'c" assumes "Sup_pres f" shows "Sup_closed_set (range f)"
proper_iff: fixes P :: "heap \<times> nat set \<Rightarrow> bool"   and as :: "nat set"   and h :: "heap"   and h' :: "heap" assumes "proper P"   and "relH as h h'"   and "in_range (h', as)" shows "P (h, as) = P (h', as)"
step2: fixes r :: "('a \<times> 'a) set"   and \<tau> :: "'a list"   and M :: "'a multiset" assumes "trans r"   and "\<tau> \<noteq> []" shows "(M \<inter>s r \<down>l \<tau>, r|\<tau>|) \<in> mul r"
retractionE: fixes g :: "'c" assumes "C.retraction g" obtains f :: "'c" where "C.ide (g \<cdot> f)"
simps_1: fixes p :: "perm"   and tset :: "('a, 'b, 'c) Tree set['a]" shows "p \<bullet> tConj tset = tConj (map_bset (permute p) tset)"
a: shows "{x \<in> local.degenerate_lotteries. \<forall>y\<in>local.degenerate_lotteries. y \<succeq>[\<R>] x} = {}"
nths_vars1': shows "nths b.dims0 b.vars1' = b.dims1"
elims_2: fixes init_nsc :: "('a \<times> 'b ns_constraint) list \<Rightarrow> 'c"   and assert_nsc :: "'a \<Rightarrow> 'c \<Rightarrow> 'a list + 'c"   and check_nsc :: "'c \<Rightarrow> 'a list + 'c"   and solution_nsc :: "'c \<Rightarrow> (nat, 'b) mapping"   and checkpoint_nsc :: "'c \<Rightarrow> 'd"   and backtrack_nsc :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and invariant_nsc :: "('a \<times> 'b ns_constraint) list \<Rightarrow> 'a set \<Rightarrow> 'c \<Rightarrow> bool"   and checked_nsc :: "('a \<times> 'b ns_constraint) list \<Rightarrow> 'a set \<Rightarrow> 'c \<Rightarrow> bool"   and to_ns :: "('a \<times> constraint) list \<Rightarrow> ('a \<times> 'b ns_constraint) list"   and from_ns :: "(nat, 'b) mapping \<Rightarrow> 'b ns_constraint list \<Rightarrow> (nat, rat) mapping"   and x :: "('a \<times> constraint) list"   and xa :: "'a set"   and xb :: "'b ns_constraint list \<times> 'c" assumes "Incremental_NS_Constraint_Ops_To_Ns_For_Incremental_Simplex init_nsc assert_nsc check_nsc solution_nsc checkpoint_nsc backtrack_nsc invariant_nsc checked_nsc to_ns from_ns"   and "Incremental_NS_Constraint_Ops_To_Ns_For_Incremental_Simplex.invariant_cs invariant_nsc to_ns x xa xb" obtains cs :: "('a \<times> constraint) list" and J :: "'a set" and ds :: "'b ns_constraint list" and s :: "'c" where "x = cs" and "xa = J" and "xb = (ds, s)" and "ds = map snd (to_ns cs) \<and> invariant_nsc (to_ns cs) J s"
unwrapA_strict: shows "unwrapA\<cdot>\<bottom> = \<bottom>"
elims_1: fixes x :: "lock_action list"   and y :: "bool" assumes "must_acquire_lock x = y" obtains "x = []" and "\<not> y"   | las :: "lock_action list" where "x = Lock # lasa__" and "y"   | las :: "lock_action list" where "x = Unlock # lasb__" and "\<not> y"   | las :: "lock_action list" where "x = UnlockFail # lasc__" and "y = must_acquire_lock lasc__"   | las :: "lock_action list" where "x = ReleaseAcquire # las" and "y = must_acquire_lock las"
rel_FGco_def: fixes L1 :: "'au \<Rightarrow> 'bh \<Rightarrow> bool"   and Co1 :: "'av \<Rightarrow> 'bi \<Rightarrow> bool"   and Co2 :: "'ax \<Rightarrow> 'bk \<Rightarrow> bool"   and Co3 :: "'ay \<Rightarrow> 'bl \<Rightarrow> bool"   and Co4 :: "'az \<Rightarrow> 'bm \<Rightarrow> bool"   and Co5 :: "'aw \<Rightarrow> 'bj \<Rightarrow> bool"   and Co6 :: "'bd \<Rightarrow> 'bp \<Rightarrow> bool"   and Contra1 :: "'ba \<Rightarrow> 'bn \<Rightarrow> bool"   and Contra2 :: "'bb \<Rightarrow> 'bo \<Rightarrow> bool"   and Contra3 :: "'be \<Rightarrow> 'bq \<Rightarrow> bool"   and Contra4 :: "'bf \<Rightarrow> 'br \<Rightarrow> bool" shows "rel_FGco L1 Co1 Co2 Co3 Co4 Co5 Co6 Contra1 Contra2 Contra3 Contra4 = rel_F L1 Co1 Co5 (rel_G Co1 Co2 Co3 Co4 Contra1 Contra2) Co3 Co6 Contra1 Contra3 Contra4"
fps_ext: fixes p :: "'b fps"   and q :: "'b fps" assumes "\<And>n. p $ n = q $ n" shows "p = q"
gb_schema_dummy_connectible1: fixes d :: "'a \<Rightarrow> nat"   and m :: "nat"   and bs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and data :: "nat \<times> nat \<times> 'd"   and D :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) set"   and f :: "('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c"   and g :: "('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c" assumes "struct_spec sel ap ab compl"   and "compl_conn compl"   and "dickson_grading d"   and "fst ` set gs \<subseteq> dgrad_p_set d m"   and "is_Groebner_basis (fst ` set gs)"   and "fst ` set bs \<subseteq> dgrad_p_set d m"   and "set ps \<subseteq> set bs \<times> (set gs \<union> set bs)"   and "unique_idx (gs @ bs) (snd data)"   and "\<And>p q. \<lbrakk>processed (p, q) (gs @ bs) ps; (p, q) \<notin>\<^sub>p D; fst p \<noteq> 0; fst q \<noteq> 0\<rbrakk> \<Longrightarrow> crit_pair_cbelow_on d m (fst ` (set gs \<union> set bs)) (fst p) (fst q)"   and "\<nexists>xs. fst (gb_schema_dummy data D bs ps) = full_gb xs"   and "f \<in> set (fst (gb_schema_dummy data D bs ps))"   and "g \<in> set (fst (gb_schema_dummy data D bs ps))"   and "(f, g) \<notin>\<^sub>p snd (gb_schema_dummy data D bs ps)"   and "fst f \<noteq> 0"   and "fst g \<noteq> 0" shows "crit_pair_cbelow_on d m (fst ` set (fst (gb_schema_dummy data D bs ps))) (fst f) (fst g)"
rel_state_state_of: fixes R :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "'f"   and b :: "'g Heap" assumes "crel_vs1 R a b" shows "local.rel_state (=) (state_of b) b"
image_f_inv_f: fixes f :: "'e \<Rightarrow> 'd"   and A :: "'d set" assumes "surj f" shows "f ` inv f ` A = A"
union_single_eq_diff: fixes x :: "'b"   and M :: "'b multiset"   and N :: "'b multiset" assumes "x, M = N" shows "M = N - (x, {#})"
similar_adm: shows "adm (\<lambda>x. fst x \<triangleleft>\<triangleright> snd x)"
normality_factorization: fixes G :: "('c, 'd) monoid_scheme"   and N :: "'c set"   and H :: "'c set" assumes "Group.group G"   and "N \<lhd> G"   and "N \<subseteq> H"   and "H \<lhd> G" shows "rcosets\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub> N \<lhd> G Mod N"
uint_le_0_iff: fixes x :: "'b word" shows "(uint x \<le> 0) = (uint x = 0)"
rel_pmf_bind_reflI: fixes p :: "'h pmf"   and R :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and f :: "'h \<Rightarrow> 'i pmf"   and g :: "'h \<Rightarrow> 'j pmf" assumes "\<And>x. x \<in> set_pmf p \<Longrightarrow> rel_pmf R (f x) (g x)" shows "rel_pmf R (p \<bind> f) (p \<bind> g)"
join_pmf_def: fixes M :: "'c pmf pmf" shows "join_pmf M = M \<bind> (\<lambda>x. x)"
wf_dual_lderiv_dual: fixes \<Sigma> :: "'a list"   and n :: "nat"   and r :: "'a atom rexp_dual"   and w :: "'a \<times> bool list" assumes "wf_dual \<Sigma> n r" shows "wf_dual \<Sigma> n (Co\<DD> \<Sigma> w r)"
iso_imp_eqpoll_carrier: fixes G :: "('b, 'c) monoid_scheme"   and H :: "('d, 'e) monoid_scheme" assumes "G \<cong> H" shows "carrier G \<approx> carrier H"
nort_null_comp: fixes p :: "ref"   and low :: "ref \<Rightarrow> ref"   and high :: "ref \<Rightarrow> ref"   and pret :: "dag"   and var :: "ref \<Rightarrow> nat"   and prebdt :: "bdt"   and repc :: "ref \<Rightarrow> ref"   and no :: "ref"   and repb :: "ref \<Rightarrow> ref"   and nort :: "dag"   and levellistb :: "ref list list"   and nb :: "nat" assumes "Dag p low high pret"   and "bdt pret var = Some prebdt"   and "Dag (repc no) (repb \<propto> low) (repb \<propto> high) nort"   and "ordered pret var"   and "wf_ll pret levellistb var"   and "nb < length levellistb"   and "\<forall>nt. nt \<notin> set (levellistb ! nb) \<longrightarrow> repb nt = repc nt"   and "\<forall>x\<in>set_of nort. var x < nb \<and> x \<in> set_of pret" shows "\<forall>x\<in>set_of nort. (repc \<propto> low) x = (repb \<propto> low) x \<and> (repc \<propto> high) x = (repb \<propto> high) x"
lex_ord_EqI: fixes xs :: "('b, 'c) oalist_ntm"   and ys :: "('b, 'd) oalist_ntm"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> order option"   and ko :: "'b nat_term_order" assumes "\<And>k. k \<in> fst ` set (fst (list_of_oalist_ntm xs)) \<union> fst ` set (fst (list_of_oalist_ntm ys)) \<Longrightarrow> f k (OAlist_lookup_ntm xs k) (OAlist_lookup_ntm ys k) = Some Eq" shows "OAlist_lex_ord_ntm ko f xs ys = Some Eq"
complete_build: fixes ps :: "(real, 'a) vec list"   and h :: "nat"   and k :: "'a list" assumes "length ps = 2 ^ h"   and "\<forall>k. distinct (map (\<lambda>p. p $ k) ps)" shows "KD_Tree.complete (build k ps)"
hyps: assumes "\<And>x. x \<in> set xs \<Longrightarrow> r2 (f y) (f x) = r1 y x" shows "map f (insort_wrt r1 y xs) = insort_wrt r2 (f y) (map f xs)"
ide_Ide: fixes a :: "'a \<times> 'a \<times> 'a" assumes "ide (fst a) \<and> VxV.ide (snd a)" shows "VxVxV.ide a"
one_generated_list_power: fixes u :: "'b list list"   and x :: "'b list" assumes "u \<in> lists {x}" shows "\<exists>k. concat u = x \<^sup>@ k"
post_meas1_bell10_fst: shows "post_meas1 2 |\<beta>\<^sub>1\<^sub>0\<rangle> 0 = - |unit_vec 4 3\<rangle>"
MkArr_Map: fixes f :: "(bool, 'arr list) concrete_category.arr" assumes "arr f" shows "MkArr (Dom f) (Cod f) (Map f) = f"
inj: shows "inj_on (\<lambda>b. (y + b) mod q) {..<q}"
empty_eq_affine_hull: fixes S :: "'c set" shows "({} = affine hull S) = (S = {})"
assms: fixes s :: "('a, 'b, 'c, 'd, 'e) system_state" assumes "reachable_state sys s" shows "I s"
elims_2: fixes D :: "ast_domain"   and ty_ent :: "'a \<Rightarrow> PDDL_STRIPS_Semantics.type option"   and x :: "'a ast_effect" assumes "ast_domain.wf_effect D ty_ent x" obtains a :: "'a atom formula list" and d :: "'a atom formula list" where "x = Effect a d" and "Ball (set a) (ast_domain.wf_fmla_atom D ty_ent) \<and> Ball (set d) (ast_domain.wf_fmla_atom D ty_ent)"
monic_prod: fixes as :: "'a set"   and f :: "'a \<Rightarrow> 'b poly" assumes "\<And>a. a \<in> as \<Longrightarrow> monic (f a)" shows "monic (prod f as)"
conseqS: fixes P :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and P' :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and Q' :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" assumes "\<turnstile>\<^sub>3\<^sub>' {P} c { Q}"   and "\<And>s n. P' (s, n) \<Longrightarrow> P (s, n)"   and "\<And>s n. Q (s, n) \<Longrightarrow> Q' (s, n)" shows "\<turnstile>\<^sub>3\<^sub>' {P'} c { Q'}"
Constrains_eq_constrains: fixes A :: "'b set"   and B :: "'b set" shows "A Co B = {F. F \<in> reachable F \<inter> A co reachable F \<inter> B}"
lang_norm: fixes S :: "nat set"   and r :: "nat rexp" shows "lang S (norm r) = lang S r"
modulo_finite_2_def: fixes x :: "finite_2"   and y :: "finite_2" shows "x mod y = (case (x, y) of (a\<^sub>1, b) \<Rightarrow> a\<^sub>1 | (a\<^sub>2, a\<^sub>1) \<Rightarrow> a\<^sub>2 | (a\<^sub>2, a\<^sub>2) \<Rightarrow> a\<^sub>1)"
norm_axis_eq: fixes i :: "'b"   and k :: "'a" shows "\<parallel>axis i k\<parallel> = \<parallel>k\<parallel>"
z1: shows "cmod (winding_number (subpath t 0 q +++ linepath (a - complex_of_real d) (a + complex_of_real e)) z) = 1"
p2_gt_0: fixes n :: "nat" shows "(0 < 2 ^ n) = (n < LENGTH('a))"
add_mset_not_empty: fixes a :: "'f"   and A :: "'f multiset" shows "add_mset a A \<noteq> {#}"
Field_Un: fixes r :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set" shows "Field (r \<union> s) = Field r \<union> Field s"
ennreal_divide_eq_0_iff: fixes a :: "ennreal"   and b :: "ennreal" shows "(a / b = 0) = (a = 0 \<or> b = \<top>)"
orbit_altdef_permutation: fixes f :: "'e \<Rightarrow> 'e"   and s :: "'e" assumes "permutation f" shows "orbit f s = {(f ^^ n) s |n. True}"
ring_1_def: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c" shows "class.ring_1 one times plus zero minus uminus \<equiv> class.monoid_mult one times \<and> class.ring plus zero minus uminus times \<and> class.zero_neq_one one zero"
long_dividesI: fixes b :: "'a list"   and r :: "'a list"   and p :: "'a list"   and q :: "'a list" assumes "b \<in> carrier (poly_ring R)"   and "r \<in> carrier (poly_ring R)"   and "p = q \<otimes>\<^bsub>poly_ring R\<^esub> b \<oplus>\<^bsub>poly_ring R\<^esub> r"   and "r = [] \<or> degree r < degree q" shows "long_divides p q (b, r)"
predE: fixes P :: "'a \<Rightarrow> bool"   and M :: "'a measure" assumes "P \<in> M \<rightarrow>\<^sub>M BOOL" shows "{x \<in> space M. P x} \<in> sets M"