hcju/isabelleps huggingface dataset:
list_all2_update_cong: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'c list"   and x :: "'b"   and y :: "'c"   and i :: "nat" assumes "list_all2 P xs ys"   and "P x y" shows "list_all2 P (xs[i := x]) (ys[i := y])"
A_def: shows "A = image_mset (normalize \<circ> primitive_part_fract) (field_poly.prime_factorization (map_poly to_fract p))"
diff_union_cancelR: fixes M :: "'c multiset"   and N :: "'c multiset" shows "M + N - N = M"
smooth_on_frechet_derivivative_comp: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'d set"   and g :: "'d \<Rightarrow> 'b"   and U :: "'d set"   and v :: "'b" assumes "smooth_on UNIV f"   and "smooth_on S g"   and "open U"   and "open UNIV"   and "g ` U \<subseteq> UNIV"   and "U \<subseteq> S" shows "smooth_on U (\<lambda>x. frechet_derivative f (at (g x)) v)"
assms_3: shows "a \<in> allAllocations N (set \<Omega>)"
convex_affine_closure_Int: fixes S :: "'b set"   and T :: "'b set" assumes "convex S"   and "affine T"   and "rel_interior S \<inter> T \<noteq> {}" shows "closure (S \<inter> T) = closure S \<inter> T"
\<G>\<F>_implies_GF: fixes \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" shows "\<forall>\<psi>\<in>\<G>\<F> \<phi> w. w \<Turnstile>\<^sub>n G\<^sub>n (F\<^sub>n \<psi>[\<F>\<G> \<phi> w]\<^sub>\<mu>)"
select_convs_2: fixes c_records :: "'d zmultiset"   and c_temp :: "'c \<Rightarrow> 'd zmultiset"   and c_msg :: "'c \<Rightarrow> 'c \<Rightarrow> 'd zmultiset list"   and c_glob :: "'c \<Rightarrow> 'd zmultiset"   and more :: "'e" shows "c_temp \<lparr>c_records = c_records, c_temp = c_temp, c_msg = c_msg, c_glob = c_glob, \<dots> = more\<rparr> = c_temp"
paval_aval_part: fixes a :: "aexp"   and s :: "char list \<Rightarrow> int" shows "paval a (part s) = aval a s"
isomorphic_to_identity_is_equivalence: fixes A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'b \<Rightarrow> 'b"   and \<phi> :: "'b \<Rightarrow> 'b" assumes "endofunctor A F"   and "natural_isomorphism A A F (identity_functor.map A) \<phi>" shows "equivalence_functor A A F"
face_of_empty: fixes S :: "'b set" shows "(S face_of {}) = (S = {})"
step: shows "\<Gamma>\<turnstile> (csxs, css, s) \<rightarrow> (cs'xs, css', t)"
lub_upper: fixes A :: "('d, 'b) tllist set"   and x :: "('d, 'b) tllist" assumes "Complete_Partial_Order.chain local.tllist_ord A"   and "x \<in> A" shows "local.tllist_ord x (local.tSup A)"
l_spanA_l_span: fixes A :: "'b set"   and H :: "'a set" assumes "ideal R A"   and "H \<subseteq> carrier M" shows "linear_span R M A (linear_span R M (carrier R) H) = linear_span R M A H"
corec_gpv_parametric': fixes S :: "'i \<Rightarrow> 'm \<Rightarrow> bool"   and A :: "'j \<Rightarrow> 'n \<Rightarrow> bool"   and C :: "'k \<Rightarrow> 'o \<Rightarrow> bool"   and R :: "'l \<Rightarrow> 'p \<Rightarrow> bool" shows "((S ===> rel_spmf (rel_generat A C (R ===> rel_sum (rel_gpv'' A C R) S))) ===> S ===> rel_gpv'' A C R) corec_gpv corec_gpv"
remdups_id_iff_distinct: fixes xs :: "'a list" shows "(remdups xs = xs) = distinct xs"
LU_def: shows "LU \<equiv> L \<union> (\<Union> {C \<in> components (T - S). \<not> bounded C} - cbox (- (b + One)) (b + One))"
bound_of_greater: fixes P :: "'a \<Rightarrow> real"   and x :: "'a" assumes "Expectations.bounded P" shows "P x \<le> bound_of P"
irreducible_m_Mp: fixes f :: "int poly" shows "irreducible_m (Mp f) = irreducible_m f"
coefficients_normh: fixes p :: "poly"   and n0 :: "nat" assumes "isnpolyh p n0" shows "\<forall>q\<in>set (coefficients p). isnpolyh q n0"
impr_bool_def: fixes a :: "bool"   and b :: "bool" shows "a r\<rightarrow> b = (a \<longrightarrow> b)"
base: shows "base \<in> {..<n} \<rightarrow> {..<p}"
code_def: fixes \<C> :: "'b list set" shows "code \<C> \<equiv> \<forall>xs ys. xs \<in> lists \<C> \<longrightarrow> ys \<in> lists \<C> \<longrightarrow> concat xs = concat ys \<longrightarrow> xs = ys"
that: fixes f :: "(nat \<Rightarrow> real) \<Rightarrow> 'n"   and g :: "'n \<Rightarrow> nat \<Rightarrow> real" assumes "homeomorphic_maps (nsphere (n - 1)) (top_of_set (sphere (0::'n) 1 \<inter> span B)) f g" shows "thesis"
devExt_shift: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and cmp :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and pin :: "'b \<Rightarrow> 'a list set"   and pout :: "'b \<Rightarrow> 'a list"   and bc :: "'b \<Rightarrow> 'a list"   and mining :: "'b \<Rightarrow> bool"   and honest :: "'a \<Rightarrow> bool"   and cb :: "nat"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and nid :: "'a"   and n' :: "nat"   and n'' :: "nat" assumes "Blockchain active cmp pin pout bc mining honest cb" shows "Blockchain.devExt active cmp bc mining t n nid (n' + n'') 0 = Blockchain.devExt active cmp bc mining t n nid n' n''"
S\<^sub>4_sound: fixes w :: "alphabet list" shows "w \<in> S\<^sub>4 \<longrightarrow> length (filter (\<lambda>x. x = a) w) = length (filter (\<lambda>x. x = b) w)"
that: fixes s :: "nat \<Rightarrow> real" assumes "s \<longlonglongrightarrow>\<^bsub>\<^esub> \<infinity>"   and "(flow0 x \<circ> s) \<longlonglongrightarrow> p"   and "\<forall>\<^sub>F n in sequentially. flow0 x (s n) \<in> {a<--<b} \<and> s n \<in> existence_ivl0 x" shows "thesis"
game2_d_def: shows "game2_d = spmf_of_set (nlists UNIV len) \<bind> (\<lambda>r. exec_gpv rf_encrypt \<A>1 Map.empty \<bind> (\<lambda>(((p0, p1), \<sigma>), s1). if valid_plain p0 \<and> valid_plain p1 then coin_spmf \<bind> (\<lambda>b. let pb = if b then p0 else p1 in spmf_of_set (nlists UNIV len) \<bind> (\<lambda>pad. let cipher = (r, pb [\<oplus>] pad) in exec_gpv (rf_encrypt_bad r) (\<A>2 cipher \<sigma>) (s1, False) \<bind> (\<lambda>(b', s2, bad). return_spmf (b' = b, bad)))) else coin_spmf \<bind> (\<lambda>b. return_spmf (b, False))))"
map_comp0: fixes g1 :: "'r \<Rightarrow> 'o"   and f1 :: "'l \<Rightarrow> 'r"   and g2 :: "'s \<Rightarrow> 'p"   and f2 :: "'m \<Rightarrow> 's"   and g3 :: "'t \<Rightarrow> 'q"   and f3 :: "'n \<Rightarrow> 't" shows "Gmap (g1 \<circ> f1) (g2 \<circ> f2) (g3 \<circ> f3) = Gmap g1 g2 g3 \<circ> Gmap f1 f2 f3"
binomial_Suc_Suc: fixes n :: "nat"   and k :: "nat" shows "\<^bsub>Suc n\<^esub>C\<^bsub>Suc k\<^esub> = \<^bsub>n\<^esub>C\<^bsub>k\<^esub> + \<^bsub>n\<^esub>C\<^bsub>Suc k\<^esub>"
sum_split2: fixes I :: "'b set"   and Q :: "'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and P :: "'b \<Rightarrow> bool" assumes "finite I" shows "(\<Sum>i | i \<in> I \<and> P i. if Q i then f i else g i) = (\<Sum>i | i \<in> I \<and> P i \<and> Q i. f i) + (\<Sum>i | i \<in> I \<and> P i \<and> \<not> Q i. g i)"
ide_char: fixes a :: "'a \<times> 'a" shows "C.VV.ide a = (C.VV.arr a \<and> C.VxV.ide a)"
apex_is_induced_by_cell: shows "tab.is_induced_by_cell (src (tab\<^sub>0 r)) (tab\<^sub>0 r) r.tab"
suminf_mono_reindex: fixes g :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'b" assumes "strict_mono g"   and "\<And>n. n \<notin> range g \<Longrightarrow> f n = (0::'b)" shows "(\<Sum>n. f (g n)) = suminf f"
eq': shows "Gauss_Jordan A $ x = Gauss_Jordan A $ y"
rec_NSTD_def: shows "rec_NSTD = Cn 1 rec_disj [Cn 1 rec_disj [Cn 1 rec_disj [Cn 1 rec_noteq [rec_stat, constn 0], Cn 1 rec_noteq [rec_left, constn 0]], Cn 1 rec_noteq [rec_right, Cn 1 rec_minus [Cn 1 rec_power [constn 2, Cn 1 rec_lg [Cn 1 rec_add [rec_right, constn 1], constn 2]], constn 1]]], Cn 1 rec_eq [rec_right, constn 0]]"
expX: shows "distributed M lborel X (\<lambda>x. ennreal (exponential_density l x))"
br_initial_def: fixes \<delta> :: "('d, 'e) ta_rule set" shows "br_initial \<delta> \<equiv> (br_iq \<delta>, br_iq \<delta>)"
inject_1: fixes x21 :: "'b"   and x22 :: "'b"   and y21 :: "'b"   and y22 :: "'b" shows "(DenyAllFromTo x21 x22 = DenyAllFromTo y21 y22) = (x21 = y21 \<and> x22 = y22)"
size_11: fixes x51 :: "'b formula"   and x52 :: "'b formula" shows "size (x51 \<^bold>\<or> x52) = size x51 + size x52 + Suc 0"
inverse_minus_eq: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "inverse (uminus a) = uminus (inverse a)"
physical_size_def: fixes ts :: "traffic" shows "physical_size ts \<equiv> fst (snd (snd (snd (snd (Rep_traffic ts)))))"
simple_ownership_distinct: fixes i :: "nat"   and j :: "nat"   and p\<^sub>i :: "'a"   and is\<^sub>i :: "instr list"   and \<theta>\<^sub>i :: "nat \<Rightarrow> nat option"   and sb\<^sub>i :: "unit"   and \<D>\<^sub>i :: "bool"   and \<O>\<^sub>i :: "nat set"   and \<R>\<^sub>i :: "nat \<Rightarrow> bool option"   and p\<^sub>j :: "'a"   and is\<^sub>j :: "instr list"   and \<theta>\<^sub>j :: "nat \<Rightarrow> nat option"   and sb\<^sub>j :: "unit"   and \<D>\<^sub>j :: "bool"   and \<O>\<^sub>j :: "nat set"   and \<R>\<^sub>j :: "nat \<Rightarrow> bool option" assumes "i < length u_ts"   and "j < length u_ts"   and "i \<noteq> j"   and "u_ts ! i = (p\<^sub>i, is\<^sub>i, \<theta>\<^sub>i, sb\<^sub>i, \<D>\<^sub>i, \<O>\<^sub>i, \<R>\<^sub>i)"   and "u_ts ! j = (p\<^sub>j, is\<^sub>j, \<theta>\<^sub>j, sb\<^sub>j, \<D>\<^sub>j, \<O>\<^sub>j, \<R>\<^sub>j)" shows "\<O>\<^sub>i \<inter> \<O>\<^sub>j = {}"
edka4_def: shows "edka4 \<equiv> let cf = c in while\<^sub>T (\<lambda>(cf, brk). \<not> brk) (\<lambda>(cf, uu_). assert (RGraph c s t cf) \<bind> (\<lambda>_. Graph.bfs cf s t \<bind> (\<lambda>p. case p of None \<Rightarrow> return (cf, True) | Some p \<Rightarrow> assert (p \<noteq> []) \<bind> (\<lambda>_. assert (Graph.isShortestPath cf s p t) \<bind> (\<lambda>_. resCap_cf_impl cf p \<bind> (\<lambda>bn. augment_cf_impl cf p bn \<bind> (\<lambda>cf. assert (RGraph c s t cf) \<bind> (\<lambda>_. return (cf, False))))))))) (cf, False) \<bind> (\<lambda>(cf, uu_). assert (RGraph c s t cf) \<bind> (\<lambda>_. Let (flow_of_cf cf) return))"
finite_set_of_finite_maps': fixes A :: "'c set"   and B :: "'d set" assumes "finite A"   and "finite B" shows "finite {m. dom m \<subseteq> A \<and> ran m \<subseteq> B}"
module_is_ag: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme" assumes "R module M" shows "aGroup M"
asm: shows "good_context s \<and> s'' = delayed_pool_write s \<and> fetch_instruction s'' = Inr v1 \<and> decode_instruction v1 = Inr v2 \<and> annul_val s'' = False \<and> supported_instruction (fst v2) = False"
0: shows "list (insert b A) = al1 @ [b] @ al2"
prodrg_ring: fixes I :: "'d set"   and A :: "'d \<Rightarrow> ('e, 'f) Ring_scheme" assumes "\<forall>k\<in>I. Ring (A k)" shows "Ring (r\<Pi>\<^bsub>I\<^esub> A)"
emp_UNpart: fixes cl :: "('test, 'atom, 'choice) cmd list"   and P :: "nat \<Rightarrow> nat set set" assumes "\<And>n. n < length cl \<Longrightarrow> {} \<notin> P n" shows "{} \<notin> UNpart cl P"
blinding_of_list_mono_pred: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and R' :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "R \<le> R'" shows "list_all2 R \<le> list_all2 R'"
fls_base_factor_to_fps_right_inverse: fixes f :: "'a fls"   and y :: "'a" shows "fls_base_factor_to_fps (fls_right_inverse f y) = fps_right_inverse (fls_base_factor_to_fps f) y"
exhaust: fixes y :: "'c DBMEntry" obtains x1 :: "'c" where "y = Le x1"   | x2 :: "'c" where "y = Lt x2"   | "y = \<infinity>"
Eq_RESUMPTION_I: fixes X :: "('f + 'g + 'h, 'i) Old_Datatype.node set \<Rightarrow> ('f + 'g + 'h, 'i) Old_Datatype.node set \<Rightarrow> bool"   and r :: "('f + 'g + 'h, 'i) Old_Datatype.node set"   and r' :: "('f + 'g + 'h, 'i) Old_Datatype.node set" assumes "X r r'"   and "\<And>r r'. X r r' \<Longrightarrow> (\<exists>a. r = TERMINAL a \<and> r' = TERMINAL a) \<or> (\<exists>R R' b. r = LINEAR b R \<and> r' = LINEAR b R' \<and> (X R R' \<or> Eq_RESUMPTION R R')) \<or> (\<exists>rs rs' c. r = BRANCH c rs \<and> r' = BRANCH c rs' \<and> (\<forall>d. X (rs d) (rs' d) \<or> Eq_RESUMPTION (rs d) (rs' d)))" shows "r = r'"
poly_rel_poly_of_list: shows "(list_all2 R ===> poly_rel) (poly_of_list_i ops) poly_of_list"
finite_character_def: fixes C :: "'b set set" shows "finite_character (C::'b set set) \<equiv> \<forall>S::'b set. (S \<in> C) = (\<forall>s\<subseteq>S. finite s \<longrightarrow> s \<in> C)"
eventually_at_split: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and "open" :: "'a set \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "class.linorder_topology less_eq less open" shows "eventually P (topological_space.at_within open x UNIV) = (eventually P (topological_space.at_within open x (ord.lessThan less x)) \<and> eventually P (topological_space.at_within open x (ord.greaterThan less x)))"
urel_integer_uminus: fixes x :: "integer"   and y :: "int" assumes "urel_integer x y" shows "urel_integer (uminus_p_integer pp x) (uminus_p p y)"
uminus2: fixes a :: "'a"   and b :: "'a" shows "ccw (0::'a) a (- b) = ccw (0::'a) b a"
map_return_pmf: fixes f :: "'e \<Rightarrow> 'd"   and x :: "'e" shows "map_pmf f (return_pmf x) = return_pmf (f x)"
ipresFreshAll_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'b"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'c"   and MOD :: "('d, 'e, 'varSort, 'f, 'g, 'var, 'b, 'c, 'h) model_scheme" shows "ipresFreshAll h hA MOD \<equiv> ipresFresh h MOD \<and> ipresFreshAbs hA MOD"
ord_defs_2: fixes compare :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "class.compare_order compare less_eq less" shows "lt_of_comp compare = less"
l10_15: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and P :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Col Bet A B C"   and "\<not> Tarski_neutral_dimensionless.Col Bet A B P" shows "\<exists>Q. Tarski_neutral_dimensionless.Perp Bet Cong A B Q C \<and> Tarski_neutral_dimensionless.OS Bet A B P Q"
q_def: fixes i :: "'a" shows "q i = lim (p i)"
Ivl_simps_2: fixes a :: "'b"   and b :: "'b" shows "upper (Ivl a b) = b"
hyps_6: fixes te :: "ExVar \<Rightarrow> Ty option"   and t :: "Ty" assumes "evm(xm \<mapsto> v2) hastyenv te "   and "te \<turnstile> em ===> t" shows "v hasty t"
trnr\<^sub>\<eta>_comp: fixes v :: "'a"   and \<mu> :: "'a"   and \<nu> :: "'a" assumes "ide v"   and "seq \<mu> \<nu>"   and "src \<mu> = trg (tab\<^sub>0 r)" shows "tab\<^sub>0.trnr\<^sub>\<eta> v (\<mu> \<cdot> \<nu>) = (\<mu> \<star> tab\<^sub>0 r) \<cdot> tab\<^sub>0.trnr\<^sub>\<eta> v \<nu>"
divide_neg_pos: 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 x zero"   and "less zero y" shows "less (divide x y) zero"
elims_2: fixes x :: "symbol list"   and xa :: "(nat \<times> symbol \<times> symbol list) list"   and xb :: "symbol list" assumes "LeftDerivation x xa xb" obtains a :: "symbol list" and b :: "symbol list" where "x = a__" and "xa = []" and "xb = b__" and "a__ = b__"   | a :: "symbol list" and d :: "nat \<times> symbol \<times> symbol list" and D :: "(nat \<times> symbol \<times> symbol list) list" and b :: "symbol list" where "x = a" and "xa = d # D" and "xb = b" and "\<exists>x. LeftDerives1 a (fst d) (snd d) x \<and> LeftDerivation x D b"
prems: fixes A :: "'a tensor" assumes "A \<in> set (A__ # As)" shows "dims A = ds"
H_1: shows "open_cover_of_open_subset Spec is_zariski_open U I V"
comp_eq_dest_lhs: fixes a :: "'g \<Rightarrow> 'f"   and b :: "'e \<Rightarrow> 'g"   and c :: "'e \<Rightarrow> 'f"   and v :: "'e" assumes "a \<circ> b = c" shows "a (b v) = c v"
connectedin_subset_topspace: fixes X :: "'a topology"   and S :: "'a set" assumes "connectedin X S" shows "S \<subseteq> topspace X"
Rep_Abs_mod: fixes m :: "int" shows "Rep_bit0 (Abs_bit0 (m mod int CARD('a bit0))) = m mod int CARD('a bit0)"
iterate_add_less_iff1: fixes j :: "nat"   and i :: "nat"   and u :: "'a"   and m :: "'a"   and n :: "'a" assumes "j \<le> i" shows "(iterate_add i u + m < iterate_add j u + n) = (iterate_add (i - j) u + m < n)"
S_inductive_set_def: fixes D :: "('c, 'd) Order_scheme" shows "S_inductive_set D = (\<forall>C. Chain D C \<longrightarrow> (\<exists>x\<in>carrier D. minimum_elem D (upper_bounds D C) x))"
assms: fixes i :: "nat" shows "E i \<in> M.events"
intros_2: fixes w :: "'b list"   and G :: "'b list set" assumes "w \<in> G" shows "w \<in> \<langle>G\<rangle>\<^sub>F"
p: shows "T\<^sub>p_on_rand' BIT s qs = 15 / 10"
substAll_def: fixes g :: "'g" shows "substAll g \<equiv> while (cond g) (step g) (uses g, phis g)"
suc_def: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and n :: "'a"   and \<sigma> :: "'b \<Rightarrow> 'c" shows "IFC_def.suc prob n \<sigma> = fst (IFC_def.step prob (n, \<sigma>))"
asprod_amult: fixes z :: "int"   and x :: "ant" assumes "0 < z" shows "z *\<^sub>a x = ant z * x"
list_upd_conv: fixes h :: "'b \<Rightarrow> 'b ref"   and p :: "'b ref"   and y :: "'b"   and q :: "'b ref" assumes "islist h p"   and "y \<notin> set (list h p)" shows "list (h(y := q)) p = list h p"
ode_d_def: fixes ode_ops :: "ode_ops" shows "ode_operations.ode_d ode_ops \<equiv> map_fun id (map_fun id (map_fun id Blinfun)) (\<lambda>n x dn d. if x \<in> ode_operations.Csafe ode_ops then ode_operations.ode_d_raw ode_ops n x dn d else (0::'e))"
no_trivial_inverse: fixes x :: "'a" assumes "x \<noteq> (0::'a)" shows "\<nexists>y. x + y = (0::'a)"
except_keys_Diff: fixes p :: "'c \<Rightarrow>\<^sub>0 'd"   and U :: "'c set" shows "except p (keys p - U) = except p (- U)"
before_r2: fixes vs :: "'a list"   and r1 :: "'a"   and r2 :: "'a" assumes "before vs r1 r2" shows "r2 \<in> set vs"
apex_is_induced_by_cell: shows "f.is_induced_by_cell (src (tab\<^sub>0 f)) (tab\<^sub>0 f) f.tab"
is_extensional: fixes f :: "'a \<times> 'a \<times> 'a" assumes "\<not> B.VVV.arr f" shows "B.\<alpha>' f = B.null"
power_mult: fixes a :: "'b list"   and m :: "nat"   and n :: "nat" shows "a \<^sup>@ (m * n) = (a \<^sup>@ m) \<^sup>@ n"
FIN: shows "SIDE_PRECOND (finite (E\<^sup>* `` {src}))"
all_le_eq_imp_eq: fixes d :: "nat"   and c :: "nat" shows "(\<forall>a. (a < d) = (a < c)) \<longrightarrow> d = c"
simps_4: fixes f :: "'a"   and v :: "'a local.term"   and va :: "'a local.term" shows "\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<Down> v \<^bold>\<star> va = \<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> v \<^bold>\<star> va"
hc: fixes x :: "'a \<Rightarrow> 'c" assumes "x \<in> carrier (DirProds Gs I)" shows "(\<lambda>j. if j \<in> J then h (g j) (x (g j)) else undefined) \<in> carrier (DirProds Hs J)"
that_1: assumes "x = 1 + \<i>\<^sub>\<int>" shows "thesis"
preserves_iso: fixes f :: "'c" assumes "C.iso f" shows "C.iso (\<I>\<^sub>C \<otimes>\<^sub>C f)"
path_independent_on_symmetricD: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set"   and B :: "'a set"   and C :: "'a set" assumes "f_range_on A f"   and "path_independent_on A f"   and "B \<subseteq> A"   and "C \<subseteq> A" shows "f (B \<union> C) = f (f B \<union> f C)"
xy_2: shows "d - a = y *\<^sub>R (b - a)"
eOp_invert: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'a, 'b) model"   and delta :: "'opSym"   and einp :: "'index \<Rightarrow> 'a withERR option"   and ebinp :: "'bindex \<Rightarrow> 'b withERR option"   and X :: "'a" assumes "eOp MOD delta einp ebinp = OK X" shows "\<exists>inp binp. einp = OKI inp \<and> ebinp = OKI binp \<and> X = igOp MOD delta inp binp \<and> igWlsInp MOD delta inp \<and> igWlsBinp MOD delta binp"
4: shows "\<forall>b. \<forall>x\<in>set xs. h \<turnstile> P x \<rightarrow>\<^sub>r True \<longrightarrow> h' \<turnstile> P x \<rightarrow>\<^sub>r b \<longrightarrow> b"
Un_Int_eq_4: fixes T :: "'c set"   and S :: "'c set" shows "T \<inter> (S \<union> T) = T"
exhaust: fixes y :: "sat_plan_variable" obtains x11 :: "nat" and x12 :: "nat" where "y = State x11 x12"   | x21 :: "nat" and x22 :: "nat" where "y = Operator x21 x22"
Fundamental_property_scd_aux: fixes prog :: "cmd" shows "BackwardSlice_wf sourcenode targetnode kind (valid_edge prog) (_Entry_) (Defs prog) (Uses prog) id (WStandardControlDependence.PDG_BS_s prog) reds (labels_nodes prog)"
\<pi>'_def: fixes Q :: "'v \<Rightarrow> enat"   and \<pi> :: "'v \<Rightarrow> 'v option"   and u :: "'v"   and v' :: "'v" shows "\<pi>' Q \<pi> u v' = (if upd_cond Q \<pi> u v' then Some u else \<pi> v')"
comp1_1: fixes f :: "nat \<Rightarrow> nat"   and g :: "nat \<Rightarrow> nat" assumes "f \<in> PrimRec1"   and "g \<in> PrimRec1" shows "(\<lambda>x. f (g x)) \<in> PrimRec1"
typeof_def: fixes v :: "v" shows "typeof v = (case v of ConstInt32 x \<Rightarrow> T_i32 | ConstInt64 x \<Rightarrow> T_i64 | ConstFloat32 x \<Rightarrow> T_f32 | ConstFloat64 x \<Rightarrow> T_f64)"
vec0_def: fixes n :: "nat" shows "vec0 n = replicate n (0::'b)"
prems_2: shows "\<^bold>|ws\<^bold>| = Suc k"
f_bounded_above: fixes c' :: "real" assumes "0 < c'" obtains c :: "real" where "\<And>x. \<lbrakk>x\<^sub>0 \<le> x; x \<le> x\<^sub>1\<rbrakk> \<Longrightarrow> f x \<le> 1 / 2 * (c * f_approx x)" and "c' \<le> c" and "0 < c"
return_time_integrable: fixes A :: "'a set" assumes "A \<in> sets MA" shows "integrable MA (\<lambda>x. real (A.return_time_function A x))"
subst_imgI: fixes \<sigma> :: "'b \<Rightarrow> ('a, 'b) Term.term"   and v :: "'b" assumes "\<sigma> v \<noteq> Var v" shows "\<sigma> v \<in> subst_range \<sigma>"
soundI': fixes SG :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list"   and G :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list" assumes "\<And>I \<nu>. \<lbrakk>is_interp I; \<And>i. \<lbrakk>0 \<le> i; i < length SG\<rbrakk> \<Longrightarrow> \<nu> \<in> seq_sem I (SG ! i)\<rbrakk> \<Longrightarrow> \<nu> \<in> seq_sem I G" shows "sound (SG, G)"
rtrancl_is_UN_relpow: fixes R :: "('a \<times> 'a) set" shows "R\<^sup>* = \<mu> (range ((^^) R))"
cnvalidD: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and n :: "nat"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and s :: "'e"   and t :: "('e, 'f) xstate" assumes "\<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P c Q,A"   and "\<forall>(P, p, Q, A)\<in>\<Theta>. \<Gamma>\<Turnstile>n:\<^bsub>/F\<^esub> P Call p Q,A"   and "\<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> =n\<Rightarrow> t"   and "s \<in> P"   and "t \<notin> Fault ` F" shows "t \<in> Normal ` Q \<union> Abrupt ` A"
arcs_ends_def: fixes G :: "('c, 'd) pre_digraph" shows "arcs_ends G \<equiv> arc_to_ends G ` arcs G"
preserves_cod: fixes f :: "'a" assumes "S.arr f" shows "S.cod (\<ll> f) = S.map (S.cod f)"
simps_4: fixes v :: "bool"   and va :: "bool list" shows "[] \<preceq>\<^sub>b v # va = False"