hcju/isabelleps huggingface dataset:
subst_typ_preserves_Some_typ_of1: fixes Ts :: "typ list"   and t :: "Core.term"   and T :: "typ"   and insts :: "((variable \<times> String.literal set) \<times> typ) list" assumes "typ_of1 Ts t = Some T" shows "typ_of1 (map (subst_typ insts) Ts) (subst_typ' insts t) = Some (subst_typ insts T)"
elims_2: fixes x :: "checker_type"   and xa :: "checker_type" assumes "checker_type_suffix x xa" obtains ts :: "t list" and ts' :: "t list" where "x = Type ts__" and "xa = Type ts'" and "suffix ts__ ts'"   | ts :: "t list" and cts :: "ct list" where "x = Type tsa__" and "xa = TopType cts__" and "ct_suffix (to_ct_list tsa__) cts__"   | cts :: "ct list" and ts :: "t list" where "x = TopType cts" and "xa = Type ts" and "ct_suffix cts (to_ct_list ts)"
ts_def: shows "ts = snd (solve_factored_ratfps' p fctrs)"
local_time_Suc: fixes A :: "'a set"   and n :: "nat"   and x :: "'a" shows "local_time A (n + 1) x = local_time A n x + indicator A ((T ^^ n) x)"
predicate2I: fixes P :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and Q :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "\<And>x y. P x y \<Longrightarrow> Q x y" shows "P \<le> Q"
tendsto_scaleR: fixes f :: "'a \<Rightarrow> real"   and a :: "real"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and b :: "'b" assumes "(f \<longlongrightarrow> a) F"   and "(g \<longlongrightarrow> b) F" shows "((\<lambda>x. f x *\<^sub>R g x) \<longlongrightarrow> a *\<^sub>R b) F"
normal_N_N1: shows "subgroup N (G\<lparr>carrier := normalizer G N1\<rparr>)"
fermat_theorem_mod_ring: fixes a :: "'b mod_ring" shows "a ^ CARD('b) = a"
that: fixes a :: "real" assumes "A \<le> a"   and "\<forall>a'\<ge>a. a' \<in> \<nat> \<longrightarrow> f \<in> O(\<lambda>x. real x powr p * (1 + integral (\<lambda>u. g' u / u powr (p + 1)) a' (real x)))" shows "thesis"
Var_altdef: fixes i :: "nat" shows "Var i = MPoly_Type.monom (Poly_Mapping.single i 1) (1::'b)"
div_normalize: 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 (normalize a) = unit_factor a"
continuous_map_eq_image_closure_subset_gen: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" shows "continuous_map X Y f = (f ` topspace X \<subseteq> topspace Y \<and> (\<forall>S. f ` (X closure_of S) \<subseteq> Y closure_of f ` S))"
implementsE: fixes jp :: "'a \<Rightarrow> ('obs, 'aAct, 'c) Protocol"   and t :: "'s Trace" assumes "implements jp"   and "t \<in> jkbpC" shows "set \<circ> actJP jp t = set \<circ> jAction MC t"
adv: fixes \<A> :: "nat \<Rightarrow> ('g + 'b, 'h + 'd) distinguisher" assumes "\<And>\<eta>. \<I>_real \<eta> \<oplus>\<^sub>\<I> \<I>_common \<eta> \<turnstile>g \<A> \<eta> \<surd>"   and "\<And>\<eta>. interaction_any_bounded_by (\<A> \<eta>) (bound_outer \<eta>)"   and "\<And>\<eta>. lossless_outer \<Longrightarrow> plossless_gpv (\<I>_real \<eta> \<oplus>\<^sub>\<I> \<I>_common \<eta>) (\<A> \<eta>)" shows "Negligible.negligible (\<lambda>\<eta>. advantage (\<A> \<eta>) (sim_outer \<eta> |\<^sub>= 1\<^sub>C \<rhd> middle \<eta>) (real \<eta>))"
height_delete_root: fixes t :: "'b tree" assumes "avl t"   and "t \<noteq> ET" shows "height t = height (delete_root t) \<or> height t = height (delete_root t) + 1"
Beta_in_subst: fixes s :: "Lambda_Z.term"   and s' :: "Lambda_Z.term"   and x :: "name"   and t :: "Lambda_Z.term" assumes "s \<rightarrow>\<^sub>\<beta> s'" shows "subst x s t \<rightarrow>\<^sub>\<beta>\<^sup>* subst x s' t"
Ln_times_Reals: fixes r :: "complex"   and z :: "complex" assumes "r \<in> \<real>"   and "0 < Re r"   and "z \<noteq> 0" shows "Ln (r * z) = complex_of_real (ln (Re r)) + Ln z"
elims_2: fixes x :: "('b ivl \<times> 'b) tree"   and xa :: "'b ivl" assumes "search x xa" obtains l :: "('b ivl \<times> 'b) tree" and a :: "'b ivl" and m :: "'b" and r :: "('b ivl \<times> 'b) tree" and x :: "'b ivl" where "xa__ = \<langle>l, (a, m), r\<rangle>" and "xa = x" and "if overlap x a then True else if l \<noteq> \<langle>\<rangle> \<and> low x \<le> max_hi l then search l x else search r x"
simps_14: fixes va :: "('b, 'c, 'd) Tree set['b]"   and v :: "('b, 'c, 'd) Tree" shows "(tConj va =\<^sub>\<alpha> tNot v) = False"
decs_evaluate_not_timeout: fixes ck :: "bool"   and mn :: "char list list"   and env :: "v sem_env"   and s :: "'b state"   and ds :: "dec list"   and r :: "'b state \<times> (v sem_env, v) result" assumes "evaluate_decs ck mn env s ds r" shows "\<forall>s' r'. ck = False \<and> r = (s', r') \<longrightarrow> r' \<noteq> Rerr (Rabort Rtimeout_error)"
g_secure_suff_def: fixes P :: "'a process"   and L :: "'a \<Rightarrow> g_level" shows "g_secure_suff P L \<equiv> \<forall>xs x. xs @ [x] \<in> traces P \<and> L x = High \<longrightarrow> {ys'. \<exists>ys. xs @ ys \<in> traces P \<and> ys' = filter (\<lambda>y. L y = Low) ys} \<subseteq> {ys'. \<exists>ys. xs @ x # ys \<in> traces P \<and> ys' = filter (\<lambda>y. L y = Low) ys}"
of_real_1: shows "of_real 1 = (1::'b)"
assms: shows "\<tau>Exec_methr ci P ins xt t h s s'"
inf_top_right: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and top :: "'c"   and x :: "'c" assumes "class.bounded_semilattice_inf_top inf less_eq less top" shows "inf x top = x"
hom_induced_carrier: fixes p :: "int"   and X :: "'c topology"   and S :: "'c set"   and Y :: "'b topology"   and T :: "'b set"   and f :: "'c \<Rightarrow> 'b"   and c :: "(((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int) set" shows "hom_induced p X S Y T f c \<in> carrier (relative_homology_group p Y T)"
dagger_slide: fixes x :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set" shows "x O (y O x)\<^sup>* = (x O y)\<^sup>* O x"
SEM_fwhile: fixes S :: "'b com"   and p :: "'b set"   and b :: "'b set"   and k :: "nat" assumes "SEM S (p \<inter> b) \<subseteq> p" shows "SEM (fwhile b S k) p \<subseteq> p \<inter> - b"
count_notin: fixes x :: "'b"   and xs :: "'b list" assumes "x \<notin> set xs" shows "count_list xs x = 0"
encode_positive_transition_frame_axiom_valid: fixes sv :: "sat_plan_variable"   and t :: "nat"   and v :: "nat \<times> nat" assumes "sv \<in> atoms (encode_positive_transition_frame_axiom (\<phi> prob_with_noop abs_prob ) t v)"   and "t < h" shows "valid_state_var sv"
get_parent_ptr_in_heap: fixes h :: "(_) heap"   and node_ptr :: "(_) node_ptr" assumes "h \<turnstile> ok get_parent node_ptr" shows "node_ptr |\<in>| node_ptr_kinds h"
rec_sigma1_def: fixes f :: "recf" shows "rec_sigma1 f = PR (CN f [CN Z [recf.Id 1 0], recf.Id 1 0]) (CN rec_add [recf.Id 3 1, CN f [CN S [recf.Id 3 0], recf.Id 3 2]])"
is_group: shows "Group.group (subgroup_generated G A \<times>\<times> subgroup_generated G B)"
axioms_1: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and smult' :: "('b, 'a) aezfun \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'c \<Rightarrow> 'd"   and W :: "'d set" assumes "FGModuleIso G smult V smult' T W" shows "FGModuleHom G smult V smult' T"
37: fixes x :: "'a"   and y :: "'a" shows "(x \<squnion> top) \<sqinter> y = y \<squnion> x \<sqinter> y"
psi_theta_1: fixes n :: "nat" shows "theta n + psi (Discrete.sqrt n) \<le> psi n"
nodes_alt_def: shows "nodes = \<Union> (reachable ` Collect init)"
tendsto_ln_powr_over_powr: fixes a :: "real"   and b :: "real" assumes "0 < a"   and "0 < b" shows "((\<lambda>x. ln x powr a / x powr b) \<longlongrightarrow> 0) at_top"
UNIV_I: fixes x :: "'d" shows "x \<in> UNIV"
max2_equals1: fixes a :: "'a"   and b :: "'a" assumes "a \<in> Field r"   and "b \<in> Field r" shows "(max2 a b = a) = ((b, a) \<in> r)"
action_loc_aux_simps_2: fixes ad' :: "'a"   and al' :: "addr_loc"   and P :: "'b prog"   and ad :: "'a"   and al :: "addr_loc"   and v :: "'a val" shows "((ad', al') \<in> action_loc_aux P (NormalAction (WriteMem ad al v))) = (ad = ad' \<and> al = al')"
FullInit_def: shows "FullInit \<equiv> BInit inp q1 mid \<and> BInit mid q2 out"
ball_divide_subset_numeral: fixes x :: "'b"   and e :: "real"   and w :: "num" shows "ball x (e / numeral w) \<subseteq> ball x e"
sum_cinner_CBasis_scaleC: fixes b :: "'c"   and f :: "'c \<Rightarrow> 'd" assumes "b \<in> CBasis" shows "(\<Sum>i\<in>CBasis. \<langle>i, b\<rangle> *\<^sub>C f i) = f b"
comp'_def: shows "comp' = (\<lambda>\<langle>ab, cd\<rangle>. [ab\<lparr>0\<^sub>\<nat>\<rparr>, cd\<lparr>1\<^sub>\<nat>\<rparr>]\<^sub>\<circ>)"
stutter_selectionD_inside: fixes s :: "nat set"   and w :: "'a llist"   and i :: "nat"   and k :: "nat" assumes "stutter_selection s w"   and "enat i < llength w"   and "enat (Suc k) < esize s"   and "nth_least s k < i"   and "i < nth_least s (Suc k)" shows "w ?! i = w ?! nth_least s k"
setWatch2_def: fixes clause :: "nat"   and literal :: "Literal"   and state :: "State" shows "setWatch2 clause literal state = state\<lparr>getWatch2 := getWatch2 state(clause \<mapsto> literal), getWatchList := (getWatchList state)(literal := clause # getWatchList state literal)\<rparr>"
one_dim_canonical_basis: fixes scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "complex \<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 dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and cinner :: "'b \<Rightarrow> 'b \<Rightarrow> complex"   and canonical_basis :: "'b list"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "class.one_dim scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner canonical_basis divide inverse one times" shows "canonical_basis = [one]"
less_finfun_def: fixes f :: "'a \<Rightarrow>f 'b"   and g :: "'a \<Rightarrow>f 'b" shows "(f < g) = (f \<le> g \<and> \<not> g \<le> f)"
bin_sc_bintr: fixes m :: "nat"   and n :: "nat"   and x :: "bool"   and w :: "int" shows "take_bit m (bin_sc n x (take_bit m w)) = take_bit m (bin_sc n x w)"
assms_3: shows "lset (ltake (enat n) (ltl xs)) \<subseteq> A"
is_top_sorted_antimono: fixes R :: "('a \<times> 'a) set"   and R' :: "('a \<times> 'a) set"   and l :: "'a list" assumes "R \<subseteq> R'"   and "is_top_sorted R' l" shows "is_top_sorted R l"
set_pmf_of_set_lessThan_length: fixes xs :: "'b list" assumes "xs \<noteq> []" shows "set_pmf (pmf_of_set {..<length xs}) = {..<length xs}"
arr_mkIde: fixes A :: "'s set" shows "arr (mkIde A) = (A \<subseteq> Univ)"
prod_rel_simp: fixes a :: "'c"   and b :: "'d"   and a' :: "'e"   and b' :: "'f"   and R1 :: "('c \<times> 'e) set"   and R2 :: "('d \<times> 'f) set" shows "(((a, b), a', b') \<in> R1 \<times>\<^sub>r R2) = ((a, a') \<in> R1 \<and> (b, b') \<in> R2)"
t': shows "t' = (p', is', \<theta>', sb', \<D>', \<O>', \<R>')"
merge_blindable_alt_def: fixes h :: "'c \<Rightarrow> 'd"   and m :: "'c \<Rightarrow> 'c \<Rightarrow> 'c option"   and x :: "('c, 'd) blindable\<^sub>m"   and y :: "('c, 'd) blindable\<^sub>m" shows "merge_blindable h m x y = (case (x, y) of (Unblinded x, Unblinded y) \<Rightarrow> map_option Unblinded (m x y) | (Unblinded x, Blinded xa) \<Rightarrow> if Content (h x) = xa then Some (Unblinded x) else None | (Blinded x, Unblinded y) \<Rightarrow> if Content (h y) = x then Some (Unblinded y) else None | (Blinded x, Blinded u) \<Rightarrow> if x = u then Some (Blinded u) else None)"
assms_2: fixes x :: "'a set" assumes "x \<in> X" shows "x \<subseteq> A"
is_none_code_2: fixes x :: "'e" shows "Option.is_none (Some x) = False"
l_inv_ex: fixes x :: "'a \<Rightarrow> 'b" assumes "x \<in> carrier (DirProds G I)" shows "\<exists>y\<in>carrier (DirProds G I). y \<otimes>\<^bsub>DirProds G I\<^esub> x = \<one>\<^bsub>DirProds G I\<^esub>"
freeInDisj: fixes var :: "nat"   and L :: "atom list" shows "freeIn var (list_disj (list_conj (map (substNegInfinity var) L) # map (elimVar var L []) L))"
lm4: fixes x :: "nat"   and key :: "nat"   and y :: "nat" assumes "c_assoc_have_key (pr_gr x) key = 0"   and "0 \<le> y" shows "c_assoc_have_key (pr_gr (x + y)) key = 0"
set_plus_mono4: fixes a :: "'a"   and C :: "'a set"   and D :: "'a set" assumes "a \<in> C" shows "a +o D \<subseteq> D + C"
u: shows "a * p - e * N * b * q = P ^ n * u"
to_single_list: fixes x :: "'a"   and xs :: "'a list" shows "x # xs = single_list x @ xs"
geodesic_segment_real: fixes G :: "real set" shows "geodesic_segment G = (\<exists>x y. x \<le> y \<and> G = {x..y})"
encode_kind_4: fixes n :: "nat"   and f :: "recf"   and g :: "recf" shows "encode_kind (encode (Pr n f g)) = 4"
bounded_cbilinear_def: fixes prod :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" shows "bounded_cbilinear prod \<equiv> ((\<forall>a a' b. prod (a + a') b = prod a b + prod a' b) \<and> (\<forall>a b b'. prod a (b + b') = prod a b + prod a b')) \<and> (\<forall>r a b. prod (r *\<^sub>C a) b = r *\<^sub>C prod a b) \<and> (\<forall>a r b. prod a (r *\<^sub>C b) = r *\<^sub>C prod a b) \<and> (\<exists>K. \<forall>a b. norm (prod a b) \<le> norm a * norm b * K)"
new_document_ptr_in_heap: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_document_ptr :: "(_) document_ptr" assumes "h \<turnstile> new_document \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new_document \<rightarrow>\<^sub>r new_document_ptr" shows "new_document_ptr |\<in>| document_ptr_kinds h'"
Ipoly_eq_insertionI: fixes p :: "'a poly"   and xs :: "nat \<Rightarrow> 'a"   and ys :: "'a list" assumes "\<And>i. i < num_params p \<Longrightarrow> xs i = ys ! i" shows "Ipoly ys p = insertion xs p"
funpow_commute: fixes k :: "nat"   and f :: "'b \<Rightarrow> 'a"   and c :: "'b \<Rightarrow> 'b"   and s :: "'b"   and c' :: "'a \<Rightarrow> 'a" assumes "\<forall>k'<k. f (c ((c ^^ k') s)) = c' (f ((c ^^ k') s))" shows "f ((c ^^ k) s) = (c' ^^ k) (f s)"
igSwapIPresIGWlsSTR_def: fixes MOD :: "('i, 'j, 'k, 'l, 'm, 'n, 'o, 'p, 'q) model_scheme" shows "igSwapIPresIGWlsSTR MOD \<equiv> \<forall>zs z1 z2 s X. igWls MOD s (igSwap MOD zs z1 z2 X) = igWls MOD s X"
heterogeneous_supterm: fixes t :: "('fun, 'var) Term.term"   and \<A> :: "('b strand_label \<times> ('fun, 'var) strand_step) list"   and Sec :: "('fun, 'var) Term.term set"   and t' :: "('fun, 'var) Term.term" assumes "heterogeneous\<^sub>l\<^sub>s\<^sub>t t \<A> Sec"   and "t \<sqsubseteq> t'" shows "heterogeneous\<^sub>l\<^sub>s\<^sub>t t' \<A> Sec"
foldD_closed: fixes B :: "'b set"   and D :: "'c set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and A :: "'b set"   and e :: "'c" assumes "LCD B D f"   and "finite A"   and "e \<in> D"   and "A \<subseteq> B" shows "foldD D f e A \<in> D"
ipPartitioning: fixes ts :: "'b set list" shows "ipPartition (set ts) (partitioning ts {})"
invertibleE: fixes u :: "'a"   and thesis :: "bool" assumes "sub.invertible u"   and "\<And>v. \<lbrakk>u \<cdot> v = \<one> \<and> v \<cdot> u = \<one>; v \<in> G\<rbrakk> \<Longrightarrow> thesis"   and "u \<in> G" shows "thesis"
x: shows "x \<in> {(last vs2, ram1), (ram1, hd vs), (last vs, ram2), (ram2, hd vs2)} \<union> Edges vs \<union> Edges vs2"
Min_ne_def: fixes X :: "'b set" shows "Min_ne X = FOLD_bin_ne min X"
edge_ex_vertices: fixes E :: "'b set set"   and e :: "'b set" assumes "Graph_E E"   and "e \<in> E" shows "\<exists>u v. u \<noteq> v \<and> e = {u, v}"
finite_group: fixes \<G> :: "'d cyclic_group" assumes "ot_base \<G>" shows "finite (carrier \<G>)"
interaction_bounded_by_exec_gpv_bad_count: fixes "consider" :: "'d \<Rightarrow> bool"   and gpv :: "('e, 'd, 'f) gpv"   and n :: "enat"   and bad :: "'g \<Rightarrow> bool"   and s :: "'g"   and callee :: "'g \<Rightarrow> 'd \<Rightarrow> ('f \<times> 'g) spmf"   and \<I> :: "('d, 'f) \<I>"   and count :: "'g \<Rightarrow> nat"   and k :: "real" assumes "interaction_bounded_by consider gpv n"   and "\<not> bad s"   and "\<And>s x y s'. \<lbrakk>(y, s') \<in> set_spmf (callee s x); consider x; x \<in> outs_\<I> \<I>\<rbrakk> \<Longrightarrow> count s' \<le> Suc (count s)"   and "\<And>s x y s'. \<lbrakk>(y, s') \<in> set_spmf (callee s x); \<not> consider x; x \<in> outs_\<I> \<I>\<rbrakk> \<Longrightarrow> count s' \<le> count s"   and "\<And>s' x. \<lbrakk>\<not> bad s'; enat (count s') < n + enat (count s); consider x; x \<in> outs_\<I> \<I>\<rbrakk> \<Longrightarrow> spmf (map_spmf (bad \<circ> snd) (callee s' x)) True \<le> k"   and "\<And>s x y s'. \<lbrakk>(y, s') \<in> set_spmf (callee s x); \<not> bad s; bad s'; x \<in> outs_\<I> \<I>\<rbrakk> \<Longrightarrow> consider x"   and "0 \<le> k"   and "\<I> \<turnstile>g gpv \<surd>"   and "\<And>s. \<I> \<turnstile>c callee s \<surd>" shows "ennreal (spmf (map_spmf (bad \<circ> snd) (exec_gpv callee gpv s)) True) \<le> ennreal k * ennreal_of_enat n"
assms: shows "x \<in> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F"
omega_sum_unfold: fixes x :: "'a"   and y :: "'a" shows "x\<^sup>\<omega> + x\<^sup>\<star> \<cdot> y \<cdot> (x + y)\<^sup>\<omega> = (x + y)\<^sup>\<omega>"
olocal_quality_increases: fixes i :: "nat"   and ANY :: "state \<Rightarrow> state \<Rightarrow> bool" shows "opaodv i \<Turnstile>\<^sub>A (act (recvmsg rreq_rrep_sn), other ANY {i} \<rightarrow>) onll \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (seqll i (\<lambda>((\<xi>, uu_), uu_, \<xi>', uu_). quality_increases \<xi> \<xi>'))"
un_img: shows "\<Union> {g - {x} |g. g \<in> G} = \<Union> G - {x}"
Compr_image_eq: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and P :: "'b \<Rightarrow> bool" shows "{x \<in> f ` A. P x} = f ` {x \<in> A. P (f x)}"
c_intersect_def: shows "c_intersect = (\<lambda>u v. set_to_nat (nat_to_set u \<inter> nat_to_set v))"
new_character_data_child_nodes: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_character_data_ptr :: "(_) character_data_ptr" assumes "h \<turnstile> new_character_data \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new_character_data \<rightarrow>\<^sub>r new_character_data_ptr" shows "h' \<turnstile> get_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a new_character_data_ptr val \<rightarrow>\<^sub>r []"
set: fixes x1 :: "'d"   and x2 :: "'d stream" shows "sset (x1 ## x2) = insert x1 (sset x2)"
linorder_sorted_wrt_list_of_set_2: fixes A :: "'b set"   and R :: "('b \<times> 'b) set"   and B :: "'b set" assumes "linorder_on A R"   and "finite B"   and "B \<subseteq> A" shows "distinct (sorted_wrt_list_of_set R B)"
is_GB_cofactor_boundE_Polys: fixes F :: "(('a option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set"   and b :: "nat"   and X :: "'a option set" assumes "extended_ord.is_GB_cofactor_bound F b"   and "F \<subseteq> P[X]" obtains G :: "(('a option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set" where "extended_ord.punit.is_Groebner_basis G" and "ideal G = ideal F" and "G \<subseteq> P[X]" and "\<And>g. g \<in> G \<Longrightarrow> \<exists>F' q. finite F' \<and> F' \<subseteq> F \<and> g = (\<Sum>f\<in>F'. q f * f) \<and> (\<forall>f. q f \<in> P[X] \<and> poly_deg (q f * f) \<le> b \<and> (f \<notin> F' \<longrightarrow> q f = 0))"
suffixes_eq_snoc: fixes ys :: "'a list"   and xs :: "'a list list"   and x :: "'a list" shows "(suffixes ys = xs @ [x]) = ((ys = [] \<and> xs = [] \<or> (\<exists>z zs. ys = z # zs \<and> xs = suffixes zs)) \<and> x = ys)"
step: fixes s2 :: "nat"   and f :: "nat \<Rightarrow> nat"   and s1 :: "nat"   and i :: "value list"   and r1 :: "nat \<Rightarrow>f value option"   and o :: "value list"   and e1 :: "((nat \<times> nat) \<times> transition) fset"   and l :: "String.literal"   and e2 :: "((nat \<times> nat) \<times> transition) fset"   and r2 :: "nat \<Rightarrow>f value option"   and es :: "(String.literal \<times> value list \<times> value list) list" assumes "s2 = f s1"   and "\<forall>(s1', t1)|\<in>|ffilter (\<lambda>(s1', t1). evaluate_outputs t1 i r1 = map Some o) (possible_steps e1 s1 r1 l i). \<exists>(s2', t2)|\<in>|possible_steps e2 s2 r2 l i. evaluate_outputs t2 i r2 = map Some o \<and> trace_simulation f e1 s1' (evaluate_updates t1 i r1) e2 s2' (evaluate_updates t2 i r2) es" shows "trace_simulation f e1 s1 r1 e2 s2 r2 ((l, i, o) # es)"
BINDINGc_def: fixes B :: "ty"   and X :: "ntrm set" shows "BINDINGc B X \<equiv> {<a>:M |a M. \<forall>x P. x:P \<in> X \<longrightarrow> SNa (M{a:=x.P})}"
update_def: fixes s :: "Var \<Rightarrow> Val"   and x :: "Var"   and v :: "Val" shows "update s x v = (\<lambda>y. if x = y then v else s y)"
ring_r_one: fixes R :: "('e, 'f) Ring_scheme"   and x :: "'e" assumes "Ring R"   and "x \<in> carrier R" shows "x \<cdot>\<^sub>r\<^bsub>R\<^esub> 1\<^sub>r\<^bsub>R\<^esub> = x"
lcm_mult_left: fixes c :: "'a"   and a :: "'a"   and b :: "'a" shows "lcm (c * a) (c * b) = normalize (c * lcm a b)"
rel_generat_inf: fixes A :: "'d \<Rightarrow> 'g \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'h \<Rightarrow> bool"   and C :: "'f \<Rightarrow> 'i \<Rightarrow> bool"   and A' :: "'d \<Rightarrow> 'g \<Rightarrow> bool"   and B' :: "'e \<Rightarrow> 'h \<Rightarrow> bool"   and C' :: "'f \<Rightarrow> 'i \<Rightarrow> bool" shows "rel_generat A B C \<sqinter> rel_generat A' B' C' = rel_generat (A \<sqinter> A') (B \<sqinter> B') (C \<sqinter> C')"
quot_of_fract_add: fixes x :: "'b fract"   and y :: "'b fract" shows "quot_of_fract (x + y) = (let (a, b) = quot_of_fract x; (c, d) = quot_of_fract y in normalize_quot (a * d + b * c, b * d))"
fset_induct_stronger: fixes P :: "'b fset \<Rightarrow> bool"   and S :: "'b fset" assumes "P {||}"   and "\<And>x S. \<lbrakk>x |\<notin>| S; P S\<rbrakk> \<Longrightarrow> P (finsert x S)" shows "P S"
rfb_surj: fixes \<phi> :: "'c set \<Rightarrow> 'd set" assumes "Inf_pres \<phi>" shows "\<exists>R. fb\<^sub>\<R> R = \<phi>"
tendsto_bot: fixes "open" :: "'a set \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and a :: "'a" assumes "class.topological_space open" shows "filterlim f (topological_space.nhds open a) bot"
app_simps_2: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<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_app unapp (free name)"
factorize_polys_irreducible_monic_2: fixes ps :: "rat poly list"   and fs :: "rat poly list"   and data :: "(rat \<times> (nat \<times> nat) list) list"   and f :: "rat poly" assumes "factorize_polys ps = (fs, data)"   and "f \<in> set fs" shows "irreducible f \<and> monic f"
g_has_sgnx1: assumes "g st = 0"   and "sin st < 0 \<or> cos st = - 1" shows "(g has_sgnx 1) (at_right st)"
swap_swap_id: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<rightleftharpoons>\<^sub>F y) ((x \<rightleftharpoons>\<^sub>F y) z) = z"
neg_numeral_le_zero: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and n :: "num" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (uminus (numeral.numeral one plus n)) zero"
f_def: shows "f = (\<lambda>n x. if x \<in> {a..real n} then P x else 0)"
p: shows "p = row_to_poly (pps_to_list S) x"